geometry actual signless distance. But in algebra it became of
importance to say that every equation of the first degree has a root,
and the notion of negative number was introduced. The negative unit had
to be defined as what can be added to the positive unit and produce the
sum zero. The corresponding notion was readily at hand in geometry,
where it was clear that a unit distance can be measured to the left or
down from the farther end of a unit distance already measured to the
right or up from a point O, with the result of reaching O again. Thus,
to give full interpretation in geometry to the algebraically negative,
it was only necessary to associate distinctness of sign with
oppositeness of direction. Later it was discovered that algebraical
reasoning would be much facilitated, and that conclusions as to the real
would retain all their soundness, if a pair of imaginary units
[+-][root]-1 of what might be called number were allowed to be
contemplated, the pair being defined, though not separately, by the two
properties of having the real sum 0 and the real product 1. Only in
these two real combinations do they enter in conclusions as to the real.
An advantage gained was that every quadratic equation, and not some
quadratics only, could be spoken of as having two roots. These
admissions of new units into algebra were final, as it admitted of proof
that all equations of degrees higher than two have the full numbers of
roots possible for their respective degrees in any case, and that every
root has a value included in the form a + b [root]-1, with a, b, real.
The corresponding enrichment could be given to geometry, with
corresponding advantages and the same absence of danger, and this was
done. On a line of measurement of distance we contemplate as existing,
not only an infinite continuum of points at real distances from an
origin of measurement O, but a doubly infinite continuum of points, all
but the singly infinite continuum of real ones imaginary, and imaginary
in conjugate pairs, a conjugate pair being at imaginary distances from
O, which have a real arithmetic and a real geometric mean. To geometry
enriched with this conception all algebra has its application.
5. Actual geometry is one, two or three-dimensional, i.e. lineal, plane
or solid. In one-dimensional geometry positions and measurements in a
single line only are admitted. Now descriptive constructions for points
in a line are impossible without going out of the line. It has therefore
been held that there is a sense in which no science of geometry strictly
confined to one dimension exists. But an algebra of one variable can be
applied to the study of distances along a line measured from a chosen
point on it, so that the idea of construction as distinct from
measurement is not essential to a one-dimensional geometry aided by
algebra. In geometry of two dimensions, the flat of the land-measurer,
the passage from one point O to any other point, can be effected by two
successive marches, one east or west and one north or south, and, as
will be seen, an algebra of two variables suffices for geometrical
exploitation. In geometry of three dimensions, that of space, any point
can be reached from a chosen one by three marches, one east or west, one
north or south, and one up or down; and we shall see that an algebra of
three variables is all that is necessary. With three dimensions actual
geometry stops; but algebra can supply any number of variables. Four or
more variables have been used in ways analogous to those in which one,
two and three variables are used for the purposes of one, two and
three-dimensional geometry, and the results have been expressed in
quasi-geometrical language on the supposition that a higher space can be
conceived of, though not realized, in which four independent directions
exist, such that no succession of marches along three of them can effect
the same displacement of a point as a march along the fourth; and
similarly for higher numbers than four. Thus analytical, though not
actual, geometries exist for four and more dimensions. They are in fact
algebras furnished with nomenclature of a geometrical cast, suggested by
convenient forms of expression which actual geometry has, in return for
benefits received, conferred on algebras of one, two and three
variables.
We will confine ourselves to the dimensions of actual geometry, and will
devote no space to the one-dimensional, except incidentally as existing
within the two-dimensional. The analytical method will now be explained
for the cases of two and three dimensions in succession. The form of it
originated by Descartes, and thence known as Cartesian, will alone be
considered in much detail.
I. _Plane Analytical Geometry._
[Illustration: FIG. 48.]
[Illustration: FIG. 49.]
6. _Coordinates._--It is assumed that the points, lines and figures
considered lie in one and the same plane, which plane therefore need
not be in any way referred to. In the plane a point O, and two lines
x'Ox, y'Oy, intersecting in O, are taken once for all, and regarded as
fixed. O is called the origin, and x'Ox, y'Oy the axes of x and y
respectively. Other positions in the plane are specified in relation
to this fixed origin and these fixed axes. From any point P we suppose
PM drawn parallel to the axis of y to meet the axis of x in M, and may
also suppose PN drawn parallel to the axis of x to meet the axis of y
in N, so that OMPN is a parallelogram. The position of P is determined
when we know OM ( = NP) and MP ( = ON). If OM is x times the unit of a
scale of measurement chosen at pleasure, and MP is y times the unit,
so that x and y have numerical values, we call x and y the (Cartesian)
coordinates of P. To distinguish them we often speak of y as the
ordinate, and of x as the abscissa.
It is necessary to attend to signs; x has one sign or the other
according as the point P is on one side or the other of the axis of y,
and y one sign or the other according as P is on one side or the other
of the axis of x. Using the letters N, E, S, W, as in a map, and
considering the plane as divided into four quadrants by the axes, the
signs are usually taken to be:
x y For quadrant
+ + N E
+ - S E
- + N W
- - S W
A point is referred to as the point (a, b), when its coordinates are x
= a, y = b. A point may be fixed, or it may be variable, i.e. be
regarded for the time being as free to move in the plane. The
coordinates (x, y) of a variable point are algebraic variables, and
are said to be "current coordinates."
The axes of x and y are usually (as in fig. 48) taken at right angles
to one another, and we then speak of them as rectangular axes, and of
x and y as "rectangular coordinates" of a point P; OMPN is then a
rectangle. Sometimes, however, it is convenient to use axes which are
oblique to one another, so that (as in fig. 49) the angle xOy between
their positive directions is some known angle [omega] distinct from a
right angle, and OMPN is always an oblique parallelogram with given
angles; and we then speak of x and y as "oblique coordinates." The
coordinates are as a rule taken to be rectangular in what follows.
7. _Equations and loci._ If (x, y) is the point P, and if we are given
that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies
at O, whatever value y may have, i.e. we are told the one fact that P
lies on the axis of y. Conversely, if P lies anywhere on the axis of
y, we have always OM = 0, i.e. x = 0. Thus the equation x = 0 is one
satisfied by the coordinates (x, y) of every point in the axis of y,
and not by those of any other point. We say that x = 0 is the equation
of the axis of y, and that the axis of y is the locus represented by
the equation x = 0. Similarly y = 0 is the equation of the axis of x.
An equation x = a, where a is a constant, expresses that P lies on a
parallel to the axis of y through a point M on the axis of x such that
OM = a. Every line parallel to the axis of y has an equation of this
form. Similarly, every line parallel to the axis of x has an equation
of the form y = b, where b is some definite constant.
These are simple cases of the fact that a single equation in the
current coordinates of a variable point (x, y) imposes one limitation
on the freedom of that point to vary. The coordinates of a point taken
at random in the plane will, as a rule, not satisfy the equation, but
infinitely many points, and in most cases infinitely many real ones,
have coordinates which do satisfy it, and these points are exactly
those which lie upon some locus of one dimension, a straight line or
more frequently a curve, which is said to be represented by the
equation. Take, for instance, the equation y = mx, where m is a given
constant. It is satisfied by the coordinates of every point P, which
is such that, in fig. 48, the distance MP, with its proper sign, is m
times the distance OM, with its proper sign, i.e. by the coordinates
of every point in the straight line through O which we arrive at by
making a line, originally coincident with x'Ox, revolve about O in the
direction opposite to that of the hands of a watch through an angle of
which m is the tangent, and by those of no other points. That line is
the locus which it represents. Take, more generally, the equation y =
[phi](x), where [phi](x) is any given non-ambiguous function of x.
Choosing any point M on x'Ox in fig. 1, and giving to x the value of
the numerical measure of OM, the equation determines a single
corresponding y, and so determines a single point P on the line
through M parallel to y'Oy. This is one point whose coordinates
satisfy the equation. Now let M move from the extreme left to the
extreme right of the line x'Ox, regarded as extended both ways as far
as we like, i.e. let x take all real values from -[oo] to [oo]. With
every value goes a point P, as above, on the parallel to y'Oy through
the corresponding M; and we thus find that there is a path from the
extreme left to the extreme right of the figure, all points P along
which are distinguished from other points by the exceptional property
of satisfying the equation by their coordinates. This path is a locus;
and the equation y = [phi](x) represents it. More generally still,
take an equation f(x, y) = 0 which involves both x and y under a
functional form. Any particular value given to x in it produces from
it an equation for the determination of a value or values of y, which
go with that value of x in specifying a point or points (x, y), of
which the coordinates satisfy the equation f(x, y) = 0. Here again, as
x takes all values, the point or points describe a path or paths,
which constitute a locus represented by the equation. Except when y
enters to the first degree only in f(x, y), it is not to be expected
that all the values of y, determined as going with a chosen value of
x, will be necessarily real; indeed it is not uncommon for all to be
imaginary for some ranges of values of x. The locus may largely
consist of continua of imaginary points; but the real parts of it
constitute a real curve or real curves. Note that we have to allow x
to admit of all imaginary, as well as of all real, values, in order to
obtain all imaginary parts of the locus.
A locus or curve may be algebraically specified in another way; viz.
we may be given two equations x = f([theta]), y = F([theta]), which
express the coordinates of any point of it as two functions of the
same variable parameter [theta] to which all values are open. As
[theta] takes all values in turn, the point (x, y) traverses the
curve.
It is a good exercise to trace a number of curves, taken as defined by
the equations which represent them. This, in simple cases, can be done
approximately by plotting the values of y given by the equation of a
curve as going with a considerable number of values of x, and
connecting the various points (x, y) thus obtained. But methods exist
for diminishing the labour of this tentative process.
Another problem, which will be more attended to here, is that of
determining the equations of curves of known interest, taken as
defined by geometrical properties. It is not a matter for surprise
that the curves which have been most and longest studied geometrically
are among those represented by equations of the simplest character.
8. _The Straight Line._--This is the simplest type of locus. Also the
simplest type of equation in x and y is Ax + By + C = 0, one of the
first degree. Here the coefficients A, B, C are constants. They are,
like the current coordinates, x, y, numerical. But, in giving
interpretation to such an equation, we must of course refer to numbers
Ax, By, C of unit magnitudes of the same kind, of units of counting
for instance, or unit lengths or unit squares. It will now be seen
that every straight line has an equation of the first degree, and that
every equation of the first degree represents a straight line.
[Illustration: FIG. 50.]
It has been seen (S 7) that lines parallel to the axes have equations
of the first degree, free from one of the variables. Take now a
straight line ABC inclined to both axes. Let it make a given angle
[alpha] with the positive direction of the axis of x, i.e. in fig. 50
let this be the angle through which Ax must be revolved
counter-clockwise about A in order to be made coincident with the
line. Let C, of coordinates (h, k), be a fixed point on the line, and
P(x, y) any other point upon it. Draw the ordinates CD, PM of C and P,
and let the parallel to the axis of x through C meet PM, produced if
necessary, in R. The right-angled triangle CRP tells us that, with the
signs appropriate to their directions attached to CR and RP,
RP = CR tan [alpha], i.e. MP - DC = (OM - OD) tan [alpha],
and this gives that
y - k = tan [alpha] (x - h),
an equation of the first degree satisfied by x and y. No point not on
the line satisfies the same equation; for the line from C to any point
off the line would make with CR some angle [beta] different from
[alpha], and the point in question would satisfy an equation y - k =
tan [beta](x - h), which is inconsistent with the above equation.
The equation of the line may also be written y = mx + b, where m = tan
[alpha], and b = k - h tan [alpha]. Here b is the value obtained for y
from the equation when 0 is put for x, i.e. it is the numerical
measure, with proper sign, of OB, the intercept made by the line on
the axis of y, measured from the origin. For different straight lines,
m and b may have any constant values we like.
Now the general equation of the first degree Ax + By + C = 0 may be
written y = -(A/B)x - C/B, unless B = 0, in which case it represents a
line parallel to the axis of y; and -A/B, -C/B are values which can be
given to m and b, so that every equation of the first degree
represents a straight line. It is important to notice that the general
equation, which in appearance contains three constants A, B, C, in
effect depends on two only, the ratios of two of them to the third. In
virtue of this last remark, we see that two distinct conditions
suffice to determine a straight line. For instance, it is easy from
the above to see that
x y
-- + -- = 1
a b
is the equation of a straight line determined by the two conditions
that it makes intercepts OA, OB on the two axes, of which a and b are
the numerical measures with proper signs: note that in fig. 50 a is
negative. Again,
y2 - y1
y - y1 = ------- (x - x1),
x2 - x1
i.e.
(y1 - y2)x - (x1 - x2)y + x1y2 - x2y1 = 0,
represents the line determined by the data that it passes through two
given points (x1, y1) and (x2, y2). To prove this find m in the
equation y - y1 = m(x - x1) of a line through (x1, y1), from the
condition that (x2, y2) lies on the line.
In this paragraph the coordinates have been assumed rectangular. Had
they been oblique, the doctrine of similar triangles would have given
the same results, except that in the forms of equation y - k = m(x -
h), y = mx + b, we should not have had m = tan [alpha].
9. _The Circle._--It is easy to write down the equation of a given
circle. Let (h, k) be its given centre C, and [rho] the numerical
measure of its given radius. Take P (x, y) any point on its
circumference, and construct the triangle CRP, in fig. 50 as above.
The fact that this is right-angled tells us that
CR^2 + RP^2 = CP^2,
and this at once gives the equation
(x - h)^2 + (y - k)^2 = [rho]^2.
A point not upon the circumference of the particular circle is at some
distance from (h, k) different from [rho], and satisfies an equation
inconsistent with this one; which accordingly represents the
circumference, or, as we say, the circle.
The equation is of the form
x^2 + y^2 + 2Ax + 2By + C = 0.
Conversely every equation of this form represents a circle: we have
only to take -A, -B, A^2 + B^2 - C for h, k, [rho]^2 respectively, to
obtain its centre and radius. But this statement must appear too
unrestricted. Ought we not to require A^2 + B^2 - C to be positive?
Certainly, if by circle we are only to mean the visible round
circumference of the geometrical definition. Yet, analytically, we
contemplate altogether imaginary circles, for which [rho]^2 is
negative, and circles, for which [rho] = 0, with all their reality
condensed into their centres. Even when [rho]^2 is positive, so that a
visible round circumference exists, we do not regard this as
constituting the whole of the circle. Giving to x any value whatever
in (x - h)^2 + (y - k)^2 = [rho]^2, we obtain two values of y, real,
coincident or imaginary, each of which goes with the abscissa x as the
ordinate of a point, real or imaginary, on what is represented by the
equation of the circle.
The doctrine of the imaginary on a circle, and in geometry generally,
is of purely algebraical inception; but it has been in its entirety
accepted by modern pure geometers, and signal success has attended the
efforts of those who, like K.G.C. von Staudt, have striven to base its
conclusions on principles not at all algebraical in form, though of
course cognate to those adopted in introducing the imaginary into
algebra.
A circle with its centre at the origin has an equation x^2 + y^2 =
[rho]^2.
In oblique coordinates the general equation of a circle is x^2 + 2xy
cos [omega] + y^2 + 2Ax + 2By + C = 0.
10. The conic sections are the next simplest loci; and it will be seen
later that they are the loci represented by equations of the second
degree. Circles are particular cases of conic sections; and they have
just been seen to have for their equations a particular class of
equations of the second degree. Another particular class of such
equations is that included in the form (Ax + By + C)(A'x + B'y + C') =
0, which represents two straight lines, because the product on the
left vanishes if, and only if, one of the two factors does, i.e. if,
and only if, (x, y) lies on one or other of two straight lines. The
condition that ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, which is often
written (a, b, c, f, g, h)(x, y, I)^2 = 0, takes this form is abc +
2fgh-af^2-bg^2 - ch^2 = 0. Note that the two lines may, in particular
cases, be parallel or coincident.
Any equation like F1(x, y) F2(x, y) ... F_n(x, y) = 0, of which the
left-hand side breaks up into factors, represents all the loci
separately represented by F1(x, y) = 0, F2(x, y) = 0, ... F_n(x, y) =
0. In particular an equation of degree n which is free from x
represents n straight lines parallel to the axis of x, and one of
degree n which is homogeneous in x and y, i.e. one which upon division
by x^n, becomes an equation in the ratio y/x, represents n straight
lines through the origin.
Curves represented by equations of the third degree are called cubic
curves. The general equation of this degree will be written (*)(x, y,
I)^3 = 0.
11. _Descriptive Geometry._--A geometrical proposition is either
descriptive or metrical: in the former case the statement of it is
independent of the idea of magnitude (length, inclination, &c.), and
in the latter it has reference to this idea. The method of coordinates
seems to be by its inception essentially metrical. Yet in dealing by
this method with descriptive propositions we are eminently free from
metrical considerations, because of our power to use general
equations, and to avoid all assumption that measurements implied are
any particular measurements.
[Illustration: FIG. 51.]
12. It is worth while to illustrate this by the instance of the
well-known theorem of the radical centre of three circles. The theorem
is that, given any three circles A, B, C (fig. 51), the common chords
[alpha][alpha]', [beta][beta]', [gamma][gamma]' of the three pairs of
circles meet in a point.
The geometrical proof is metrical throughout:--
Take O the point of intersection of [alpha][alpha]', [beta][beta]',
and joining this with [gamma]', suppose that [gamma]'O does not pass
through [gamma], but that it meets the circles A, B in two distinct
points [gamma]2, [gamma]1 respectively. We have then the known
metrical property of intersecting chords of a circle; viz. in circle
C, where [alpha][alpha]', [beta][beta]', are chords meeting at a point
O,
O[alpha].O[alpha]' = O[beta].O[beta]',
where, as well as in what immediately follows, O[alpha], &c. denote,
of course, _lengths_ or _distances_.
Similarly in circle A,
O[beta].O[beta]' = O[gamma]2.O[gamma]',
and in circle B,
O[alpha].O[alpha]' = O[gamma]1.O[gamma]'.
Consequently O[gamma]1.O[gamma]' = O[gamma]2.O[gamma]', that is,
O[gamma]1 = O[gamma]2, or the points [gamma]1 and [gamma]2 coincide;
that is, they each coincide with [gamma].
We contrast this with the analytical method:--
Here it only requires to be known that an equation Ax + By + C = 0
represents a line, and an equation x^2 + y^2 + Ax + By + C = 0
represents a circle. A, B, C have, in the two cases respectively,
metrical significations; but these we are not concerned with. Using S
to denote the function x^2 + y^2 + Ax + By + C, the equation of a
circle is S = o. Let the equation of any other circle be S', = x^2 +
y^2 + A'x + B'y + C' = 0; the equation S - S' = 0 is a linear equation
(S - S' is in fact = (A - A')x + (B - B')y + C - C), and it thus
represents a line; this equation is satisfied by the coordinates of
each of the points of intersection of the two circles (for at each of
these points S = 0 and S' = 0, therefore also S - S' = 0); hence the
equation S - S' = 0 is that of the line joining the two points of
intersection of the two circles, or say it is the equation of the
common chord of the two circles. Considering then a third circle S", =
x^2 + y^2 + A"x + B"y + C" = 0, the equations of the common chords are
S-S' = 0, S - S" = 0, S' - S" = 0 (each of these a linear equation);
at the intersection of the first and second of these lines S = S' and
S = S", therefore also S' = S", or the equation of the third line is
satisfied by the coordinates of the point in question; that is, the
three chords intersect in a point O, the coordinates of which are
determined by the equations S = S' = S".
It further appears that if the two circles S = 0, S' = 0 do not
intersect in any real points, they must be regarded as intersecting in
two imaginary points, such that the line joining them is the real line
represented by the equation S - S' = 0; or that two circles, whether
their intersections be real or imaginary, have always a real common
chord (or radical axis), and that for _any_ three circles the common
chords intersect in a point (of course real) which is the radical
centre. And by this very theorem, given two circles with imaginary
intersections, we can, by drawing circles which meet each of them in
real points, construct the radical axis of the first-mentioned two
circles.
13. The principle employed in showing that the equation of the common
chord of two circles is S - S' = 0 is one of very extensive
application, and some more illustrations of it may be given.
Suppose S = 0, S' = 0 are lines (that is, let S, S' now denote linear
functions Ax + By + C, A'x + B'y + C'), then S - kS' = 0 (k an
arbitrary constant) is the equation of any line passing through the
point of intersection of the two given lines. Such a line may be made
to pass through any given point, say the point (x0, y0); if S0, S'0
are what S, S' respectively become on writing for (x, y) the values
(x0, y0), then the value of k is k = S0 : S'0. The equation in fact is
SS'0 - S0S' = 0; and starting from this equation we at once verify it
_a posteriori_; the equation is a linear equation satisfied by the
values of (x, y) which make S = 0, S' = 0; and satisfied also by the
values (x0, y0); and it is thus the equation of the line in question.
If, as before, S = 0, S' = 0 represent circles, then (k being
arbitrary) S - kS' = 0 is the equation of any circle passing through
the two points of intersection of the two circles; and to make this
pass through a given point (x0, y0) we have again k = S0 : S'0. In the
particular case k = 1, the circle becomes the common chord (more
accurately it becomes the common chord together with the line
infinity; see S 23 below).
If S denote the general quadric function,
S = ax^2 +2hxy + by^2 + 2fy + 2gx + c,
then the equation S = 0 represents a conic; assuming this, then, if S'
= 0 represents another conic, the equation S - kS' = 0 represents
_any_ conic through the four points of intersection of the two conics.
[Illustration: FIG. 52.]
14. The object still being to illustrate the mode of working with
coordinates for descriptive purposes, we consider the theorem of the
polar of a point in regard to a circle. Given a circle and a point O
(fig. 52), we draw through O any two lines meeting the circle in the
points A, A' and B, B' respectively, and then taking Q as the
intersection of the lines AB' and A'B, the theorem is that the locus
of the point Q is a right line depending only upon O and the circle,
but independent of the particular lines OAA' and OBB'.
Taking O as the origin, and for the axes any two lines through O at
right angles to each other, the equation of the circle will be
x^2 + y^2 + 2Ax + 2By + C = 0;
and if the equation of the line OAA' is taken to be y = mx, then the
points A, A' are found as the intersections of the straight line with
the circle; or to determine x we have
x^2(1 + m^2) + 2x(A + Bm) + C = 0.
If(x1, y1) are the coordinates of A, and (x2, y2) of A', then the
roots of this equation are x1, x2, whence easily
1 1 A + Bm
-- + -- = -2 ------.
x1 x2 C
And similarly, if the equation of the line OBB' is taken to be y =
m'x1 and the coordinates of B, B' to be (x3, y3) and (x4, y4)
respectively, then
1 1 A + Bm'
-- + -- = -2 -------.
x3 x4 C'
We have then by S 8
x(y1 - y4) - y(x1 - x4) + x1y4 - x4y1 = 0,
x(y2 - y3) - y(x2 - x3) + x2y3 - x3y2 = 0,
as the equations of the lines AB' and A'B respectively. Reducing by
means of the relations y1 - mx1 = 0, y2 - mx2 = 0, y3 - m'x3 = 0, y4 -
m'x4 = 0, the two equations become
x(mx1 - m'x4) - y(x1 - x4) + (m'- m)x1x4 = 0,
x(mx2 - m'x3) - y(x2 - x3) + (m'- m)x2x3 = 0,
and if we divide the first of these equations by x1x4, and the second
by x2x3 and then add, we obtain
_ _ _ _
| / 1 1 \ / 1 1 \ | | 1 1 / 1 1 \ |
x| m( -- + -- ) - m'( -- + -- ) | - y| -- + -- - ( -- + -- ) |
|_ \ x3 x4 / \ x1 x2/ _| |_ x3 x4 \ x1 x2/ _|
+ 2m' - 2m = 0,
or, what is the same thing,
/ 1 1 \ / 1 1 \
( -- + -- )(y - m'x) - ( -- + -- )(y - mx) + 2m' - 2m = 0,
\ x1 x2 / \ x3 x4 /
which by what precedes is the equation of a line through the point Q.
Substituting herein for 1/x1 + 1/x2, 1/x3 + 1/x4 their foregoing
values, the equation becomes
-(A + Bm)(y - m'x) + (A + Bm')(y - mx) + C(m' - m) = 0;
that is,
(m - m')(Ax + By + C) = 0;
or finally it is Ax + By + C = 0, showing that the point Q lies in a
line the position of which is independent of the particular lines
OAA', OBB' used in the construction. It is proper to notice that there
is no correspondence to each other of the points A, A' and B, B'; the
grouping might as well have been A, A' and B', B; and it thence
appears that the line Ax + By + C = 0 just obtained is in fact the
line joining the point Q with the point R which is the intersection of
AB and A'B'.
15. In S 8 it has been seen that two conditions determine the equation
of a straight line, because in Ax + By + C = 0 one of the coefficients
may be divided out, leaving only two parameters to be determined.
Similarly five conditions instead of six determine an equation of the
second degree (a, b, c, f, g, h)(x, y, 1)^2 = 0, and nine instead of
ten determine a cubic (*)(x, y, 1)^3 = 0. It thus appears that a cubic
can be made to pass through 9 given points, and that the cubic so
passing through 9 given points is completely determined. There is,
however, a remarkable exception. Considering two given cubic curves S
= 0, S' = 0, these intersect in 9 points, and through these 9 points
we have the whole series of cubics S - kS' = 0, where k is an
arbitrary constant: k may be determined so that the cubic shall pass
through a given tenth point (k = S0 : S'0, if the coordinates are (x0,
y0), and S0, S'0 denote the corresponding values of S, S'). The
resulting curve SS'0 - S'S0 = 0 may be regarded as the cubic
determined by the conditions of passing through 8 of the 9 points and
through the given point (x0, y0); and from the equation it thence
appears that the curve passes through the remaining one of the 9
points. In other words, we thus have the theorem, any cubic curve
which passes through 8 of the 9 intersections of two given cubic
curves passes through the 9th intersection.
The applications of this theorem are very numerous; for instance, we
derive from it Pascal's theorem of the inscribed hexagon. Consider a
hexagon inscribed in a conic. The three alternate sides constitute a
cubic, and the other three alternate sides another cubic. The cubics
intersect in 9 points, being the 6 vertices of the hexagon, and the 3
Pascalian points, or intersections of the pairs of opposite sides of
the hexagon. Drawing a line through two of the Pascalian points, the
conic and this line constitute a cubic passing through 8 of the 9
points of intersection, and it therefore passes through the remaining
point of intersection--that is, the third Pascalian point; and since
obviously this does not lie on the conic, it must lie on the
line--that is, we have the theorem that the three Pascalian points (or
points of intersection of the pairs of opposite sides) lie on a line.
16. _Metrical Theory resumed. Projections and Perpendiculars._--It
is a metrical fact of fundamental importance, already used in S 8,
that, if a finite line PQ be projected on any other line OO' by
perpendiculars PP', QQ' to OO', the length of the projection P'Q' is
equal to that of PQ multiplied by the cosine of the acute angle
between the two lines. Also the algebraical sum of the projections of
the sides of any closed polygon upon any line is zero, because as a
point goes round the polygon, from any vertex A to A again, the point
which is its projection on the line passes from A' the projection of A
to A' again, i.e. traverses equal distances along the line in positive
and negative senses. If we consider the polygon as consisting of two
broken lines, each extending from the same initial to the same
terminal point, the sum of the projections of the lines which compose
the one is equal, in sign and magnitude, to the sum of the projections
of the lines composing the other. Observe that the projection on a
line of a length perpendicular to the line is zero.
Let us hence find the equation of a straight line such that the
perpendicular OD on it from the origin is of length [rho] taken as
positive, and is inclined to the axis of x at an angle xOD = [alpha],
measured counter-clockwise from Ox. Take any point P (x, y) on the
line, and construct OM and MP as in fig. 48. The sum of the
projections of OM and MP on OD is OD itself; and this gives the
equation of the line
x cos [alpha] + y sin [alpha] = [rho].
Observe that cos [alpha] and sin [alpha] here are the sin [alpha] and
-cos [alpha], or the -sin [alpha] and cos [alpha] of S 8 according to
circumstances.
We can write down an expression for the perpendicular distance from
this line of any point (x', y') which does not lie upon it. If the
parallel through (x', y') to the line meet OD in E, we have x' cos
[alpha] + y' sin [alpha] = OE, and the perpendicular distance required
is OD - OE, i.e. [rho] - x' cos [alpha] - y' sin [alpha]; it is the
perpendicular distance taken positively or negatively according as
(x', y') lies on the same side of the line as the origin or not.
The general equation Ax + By + C = 0 may be given the form x cos
[alpha] + y sin [alpha] - [rho] = 0 by dividing it by [root](A^2 +
B^3). Thus (Ax' + By' + C) / [root](A^2 + B^2) is in absolute value
the perpendicular distance of (x', y') from the line Ax + By + C = 0.
Remember, however, that there is an essential ambiguity of sign
attached to a square root. The expression found gives the distance
taken positively when (x', y') is on the origin side of the line, if
the sign of C is given to [root](A^2 + B^2).
17. _Transformation of Coordinates._--We often need to adopt new axes
of reference in place of old ones; and the above principle of
projections readily expresses the old coordinates of any point in
terms of the new.
[Illustration: FIG. 53.]
Suppose, for instance, that we want to take for new origin the point
O' of old coordinates OA = h, AO' = k, and for new axes of X and Y
lines through O' obtained by rotating parallels to the old axes of x
and y through an angle [theta] counter-clockwise. Construct (fig. 53)
the old and new coordinates of any point P. Expressing that the
projections, first on the old axis of x and secondly on the old axis
of y, of OP are equal to the sums of the projections, on those axes
respectively, of the parts of the broken line OO'M'P, we obtain:
x = h + X cos [theta] + Y cos ([theta] + 1/2[pi]) = h + X cos [theta] -
Y sin [theta],
and
y = k + X cos (1/2[pi] - [theta]) + Y cos [theta] = k + X sin [theta] +
Y cos [theta].
Be careful to observe that these formulae do not apply to every
conceivable change of reference from one set of rectangular axes to
another. It might have been required to take O'X, O'Y' for the
positive directions of the new axes, so that the change of directions
of the axes could not be effected by rotation. We must then write -Y
for Y in the above.
Were the new axes oblique, making angles [alpha], [beta] respectively
with the old axis of x, and so inclined at the angle [beta] - [alpha],
the same method would give the formulae
x = h + X cos [alpha] + Y cos [beta], y = k + X sin [alpha] + Y sin [beta].
18. _The Conic Sections._--The conics, as they are now called, were at
first defined as curves of intersection of planes and a cone; but
Apollonius substituted a definition free from reference to space of
three dimensions. This, in effect, is that a conic is the locus of a
point the distance of which from a given point, called the focus, has
a given ratio to its distance from a given line, called the directrix
(see CONIC SECTION). If e : 1 is the ratio, e is called the
eccentricity. The distances are considered signless.
Take (h, k) for the focus, and x cos [alpha] + y sin [alpha] - p = 0
for the directrix. The absolute values of [root] {(x - h)^2 + (y -
k)^2} and p - x cos [alpha] -y sin [alpha] are to have the ratio e :
1; and this gives
(x - h)^2 + (y - k)^2 = e^2(p - x cos [alpha] - y sin [alpha])^2
as the general equation, in rectangular coordinates, of a conic.
It is of the second degree, and is the general equation of that
degree. If, in fact, we multiply it by an unknown [lambda], we can, by
solving six simultaneous equations in the six unknowns [lambda], h, k,
e, p, [alpha], so choose values for these as to make the coefficients
in the equation equal to those in any equation of the second degree
which may be given. There is no failure of this statement in the
special case when the given equation represents two straight lines, as
in S 10, but there is speciality: if the two lines intersect, the
intersection and either bisector of the angle between them are a focus
and directrix; if they are united in one line, any point on the line
and a perpendicular to it through the point are: if they are parallel,
the case is a limiting one in which e and h^2 + k^2 have become
infinite while e^(-2)(h^2 + k^2) remains finite. In the case (S 9) of
an equation such as represents a circle there is another instance of
proceeding to a limit: e has to become 0, while ep remains finite:
moreover [alpha] is indeterminate. The centre of a circle is its
focus, and its directrix has gone to infinity, having no special
direction. This last fact illustrates the necessity, which is also
forced on plane geometry by three-dimensional considerations, of
treating all points at infinity in a plane as lying on a single
straight line.
Sometimes, in reducing an equation to the above focus and directrix
form, we find for h, k, e, p, tan [alpha], or some of them, only
imaginary values, as quadratic equations have to be solved; and we
have in fact to contemplate the existence of entirely imaginary
conics. For instance, no real values of x and y satisfy x^2 + 2y^2 + 3
= 0. Even when the locus represented is real, we obtain, as a rule,
four sets of values of h, k, e, p, of which two sets are imaginary; a
real conic has, besides two real foci and corresponding directrices,
two others that are imaginary.
In oblique as well as rectangular coordinates equations of the second
degree represent conics.
19. _The three Species of Conics._--A real conic, which does not
degenerate into straight lines, is called an ellipse, parabola or
hyperbola according as e <, = , or > 1. To trace the three forms it is best so to
choose the axes of reference as to simplify their equations.
In the case of a parabola, let 2c be the distance between the given
focus and directrix, and take axes referred to which these are the
point (c, 0) and the line x = - c. The equation becomes (x - c)^2 +
y^2 = (x + c)^2, i.e. y^2 = 4cx.
In the other cases, take a such that a(e ~ e^(-1)) is the distance of
focus from directrix, and so choose axes that these are (ae, 0) and x
= ae^(-1), thus getting the equation(x - ae)^2 + y^2 = e^2(x -
ae^(-1))^2, i.e. (1 - e^2)x^2 + y^2 = a^2(1 - e^2). When e < 1, i.e.
in the case of an ellipse, this may be written x^2/a^2 + y^2/b^2 = 1,
where b^2 = a^2(1 - e^2); and when e > 1, i.e. in the case of an
hyperbola, x^2/a^2 - y^2/b^2 = 1, where b^2 = a^2(e^2 - 1). The axes
thus chosen for the ellipse and hyperbola are called the principal
axes.
In figs. 54, 55, 56 in order, conics of the three species, thus
referred, are depicted.
[Illustration: FIG. 54]
[Illustration: FIG. 55]
[Illustration: FIG. 56.]
The oblique straight lines in fig. 56 are the _asymptotes_ x/a =
[+-]y/b of the hyperbola, lines to which the curve tends with
unlimited closeness as it goes to infinity. The hyperbola would have
an equation of the form xy = c if referred to its asymptotes as axes,
the coordinates being then oblique, unless a = b, in which case the
hyperbola is called rectangular. An ellipse has two imaginary
asymptotes. In particular a circle x^2 + y^2 = a^2, a particular
ellipse, has for asymptotes the imaginary lines x = [+-]y [root]-1.
These run from the centre to the so-called circular points at
infinity.
20. _Tangents and Curvature._--Let (x', y') and (x' + h, y' + k) be
two neighbouring points P, P' on a curve. The equation of the line on
which both lie is h(y - y') = k(x - x'). Now keep P fixed, and let P'
move towards coincidence with it along the curve. The connecting line
will tend towards a limiting position, to which it can never attain as
long as P and P' are distinct. The line which occupies this limiting
position is the tangent at P. Now if we subtract the equation of the
curve, with (x', y') for the coordinates in it, from the like equation
in (x' + h, y' + k), we obtain a relation in h and k, which will, as a
rule, be of the form 0 = Ah + Bk + terms of higher degrees in h and k,
where A, B and the other coefficients involve x' and y'. This gives
k/h = -A/B + terms which tend to vanish as h and k do, so that -A : B
is the limiting value tended to by k : h. Hence the equation of the
tangent is B(y - y') + A(x - x') = 0.
The _normal_ at (x', y') is the line through it at right angles to the
tangent, and its equation is A(y - y') - B(x - x') = 0.
In the case of the conic (a, b, c, f, g, h) (x, y, 1)^2 = 0 we find
that A/B = (ax' + hy' + g)/(hx' + by' + f).
We can obtain the coordinates of Q, the intersection of the normals
QP, QP' at (x', y') and (x' + h, y' + k), and then, using the limiting
value of k : h, deduce those of its limiting position as P' moves up
to P. This is the _centre of curvature_ of the curve at P (x', y'),
and is so called because it is the centre of the circle of closest
contact with the curve at that point. That it is so follows from the
facts that the closest circle is the limit tended to by the circle
which touches the curve at P and passes through P', and that the arc
from P to P' of this circle lies between the circles of centre Q and
radii QP, QP', which circles tend, not to different limits as P' moves
up to P, but to one. The distance from P to the centre of curvature is
the _radius of curvature_.
21. _Differential Plane Geometry._--The language and notation of the
differential calculus are very useful in the study of tangents and
curvature. Denoting by ([xi], [eta]) the current coordinates, we find,
as above, that the tangent at a point (x, y) of a curve is [eta] - y =
([xi] - x)dy/dx, where dy/dx is found from the equation of the curve.
If this be f(x, y) = 0 the tangent is ([xi] - x) (dPf/dPx) + ([eta] -
y) (dPf/dPy) = 0. If [rho] and ([alpha], [beta]) are the radius and
centre of curvature at (x, y), we find that q([alpha] - x) = -p(1 +
p^2), q([beta] - y) = 1 + p^2, q^2[rho]^2 = (1 + p^2)^3, where p, q
denote dy/dx, d^2y/dx^2 respectively. (See INFINITESIMAL CALCULUS.)
In any given case we can, at all events in theory, eliminate x, y
between the above equations for [alpha] - x and [beta] - y, and the
equation of the curve. The resulting equation in ([alpha], [beta])
represents the locus of the centre of curvature. This is the _evolute_
of the curve.
22. _Polar Coordinates._--In plane geometry the distance of any point
P from a fixed origin (or pole) O, and the inclination xOP of OP to a
fixed line Ox, determine the point: r, the numerical measure of OP,
the _radius vector_, and [theta], the circular measure of xOP, the
_inclination_, are called polar coordinates of P. The formulae x = r
cos [theta], y = r sin [theta] connect Cartesian and polar
coordinates, and make transition from either system to the other easy.
In polar coordinates the equations of a circle through O, and of a
conic with O as focus, take the simple forms r = 2a cos
([theta]-[alpha]), r {1 - e cos ([theta]-[alpha])} = l. The use of
polar coordinates is very convenient in discussing curves which have
properties of symmetry akin to that of a regular polygon, such curves
for instance as r = a cos m [theta], with m integral, and also the
curves called spirals, which have equations giving r as functions of
[theta] itself, and not merely of sin [theta] and cos [theta]. In the
geometry of motion under central forces the advantage of working with
polar coordinates is great.
23. _Trilinear and Areal Coordinates._--Consider a fixed triangle ABC,
and regard its sides as produced without limit. Denote, as in
trigonometry, by a, b, c the positive numbers of units of a chosen
scale contained in the lengths BC, CA, AB, by A, B, C the angles, and
by [Delta] the area, of the triangle. We might, as in S 6, take CA, CB
as axes of x and y, inclined at an angle C. Any point P (x, y) in the
plane is at perpendicular distances y sin C and x sin C from CA and
CB. Call these [beta] and [alpha] respectively. The signs of [beta]
and [alpha] are those of y and x, i.e. [beta] is positive or negative
according as P lies on the same side of CA as B does or the opposite,
and similarly for [alpha]. An equation in (x, y) of any degree may,
upon replacing in it x and y by [alpha] cosec C and [beta] cosec C, be
written as one of the same degree in ([alpha], [beta]). Now let
[gamma] be the perpendicular distance of P from the third side AB,
taken as positive or negative as P is on the C side of AB or not. The
geometry of the figure tells us that a[alpha] + b[beta] + c[gamma] =
2[Delta]. By means of this relation in [alpha], [beta], [gamma] we can
give an equation considered countless other forms, involving two or
all of [alpha], [beta], [gamma]. In particular we may make it
_homogeneous_ in [alpha], [beta], [gamma]: to do this we have only to
multiply the terms of every degree less than the highest present in
the equation by a power of (a[alpha] + b[beta] + c[gamma])/2[Delta]
just sufficient to raise them, in each case, to the highest degree.
We call ([alpha], [beta], [gamma]) _trilinear coordinates_, and an
equation in them the trilinear equation of the locus represented.
Trilinear equations are, as a rule, dealt with in their homogeneous
forms. An advantage thus gained is that we need not mean by ([alpha],
[beta], [gamma]) the actual measures of the perpendicular distances,
but any properly signed numbers which have the same ratio two and two
as these distances.
In place of [alpha], [beta], [gamma] it is lawful to use, as
coordinates specifying the position of a point in the plane of a
triangle of reference ABC, any given multiples of these. For instance,
we may use x = a[alpha]/2[Delta], y = b[beta]/2[Delta], z =
c[gamma]/2[Delta], the properly signed ratios of the triangular areas
PBC, PCA, PAB to the triangular area ABC. These are called the _areal_
coordinates of P. In areal coordinates the relation which enables us
to make any equation homogeneous takes the simple form x + y + z = 1;
and, as before, we need mean by x, y, z, in a homogeneous equation,
only signed numbers in the right ratios.
Straight lines and conics are represented in trilinear and in areal,
because in Cartesian, coordinates by equations of the first and second
degrees respectively, and these degrees are preserved when the
equations are made homogeneous. What must be said about points
infinitely far off in order to make universal the statement, to which
there is no exception as long as finite distances alone are
considered, that _every_ homogeneous equation of the first degree
represents a straight line? Let the point of areal coordinates (x',
y', z') move infinitely far off, and mean by x, y, z finite quantities
in the ratios which x', y', z' tend to assume as they become infinite.
The relation x' + y' + z' = 1 gives that the limiting state of things
tended to is expressed by x + y + z = 0. This particular equation of
the first degree is satisfied by no point at a finite distance; but we
see the propriety of saying that it has to be taken as satisfied by
all the points conceived of as actually at infinity. Accordingly the
special property of these points is expressed by saying that they lie
on a special straight line, of which the areal equation is x + y + z =
0. In trilinear coordinates this _line at infinity_ has for equation
a[alpha] + b[beta] + c[gamma] = 0.
On the one special line at infinity parallel lines are treated as
meeting. There are on it two special (imaginary) points, the circular
points at infinity of S 19, through which all circles pass in the same
sense. In fact if S = O be one circle, in areal coordinates, S + (x +
y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any
other; since the added terms are once lx + my + nz, and have the
generality of any expression like a'x + b'y + c' in Cartesian
coordinates. Now these two circles intersect in the two points where
either meets x + y + z = 0 as well as in two points on the radical
axis lx + my + nz = 0.
24. Let us consider the perpendicular distance of a point ([alpha]',
[beta]', [gamma]') from a line l[alpha] + m[beta] + n[gamma]. We can
take rectangular axes of Cartesian coordinates (for clearness as to
equalities of angle it is best to choose an origin inside ABC), and
refer to them, by putting expressions p - x cos[theta] - y sin[theta],
&c., for [alpha] &c.; we can then apply S 16 to get the perpendicular
distance; and finally revert to the trilinear notation. The result is
to find that the required distance is
(l[alpha]' + m[beta]' + n[gamma]')/{l, m, n},
where {l, m, n}^2 = l^2 + m^2 + n^2 - 2mn cos A - 2nl cos B - 2lm cos C.
In areal coordinates the perpendicular distance from (x', y', z') to
lx + my + nz = 0 is 2[Delta](lx' + my' + nz')/{al, bm, cn}. In both
cases the coordinates are of course actual values.
Now let [xi], [eta], [zeta] be the perpendiculars on the line from the
vertices A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1),
with signs in accord with a convention that oppositeness of sign
implies distinction between one side of the line and the other. Three
applications of the result above give
[xi]/l = 2[Delta]/{al, bm, cn} = [eta]/m = [zeta]/n;
and we thus have the important fact that [xi]x' + [eta]y' + [zeta]z'
is the perpendicular distance between a point of areal coordinates
(x'y'z') and a line on which the perpendiculars from A, B, C are [xi],
[eta], [zeta] respectively. We have also that [xi]x + [eta]y + [zeta]z
= 0 is the areal equation of the line on which the perpendiculars are
[xi], [eta], [zeta]; and, by equating the two expressions for the
perpendiculars from (x', y', z') on the line, that in all cases
{a[xi], b[eta], c[zeta]}^2 = 4[Delta]^2.
25. _Line-coordinates. Duality._--A quite different order of ideas
may be followed in applying analysis to geometry. The notion of a
straight line specified may precede that of a point, and points may be
dealt with as the intersections of lines. The specification of a line
may be by means of coordinates, and that of a point by an equation,
satisfied by the coordinates of lines which pass through it. Systems
of _line-coordinates_ will here be only briefly considered. Every such
system is allied to some system of point-coordinates; and space will
be saved by giving prominence to this fact, and not recommencing _ab
initio_.
Suppose that any particular system of point-coordinates, in which lx +
my + nz = 0 may represent any straight line, is before us: notice that
not only are trilinear and areal coordinates such systems, but
Cartesian coordinates also, since we may write x/z, y/z for the
Cartesian x, y, and multiply through by z. The line is exactly
assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n)
the _coordinates_ of the line. Now keep x, y, z constant, and let the
coordinates of the line vary, but always so as to satisfy the
equation. This equation, which we now write xl + ym + zn = 0, is
satisfied by the coordinates of every line through a certain fixed
point, and by those of no other line; it is the equation of that point
in the line-coordinates l, m, n.
Line-coordinates are also called _tangential_ coordinates. A curve is
the envelope of lines which touch it, as well as the locus of points
which lie on it. A homogeneous equation of degree above the first in
l, m, n is a relation connecting the coordinates of every line which
touches some curve, and represents that curve, regarded as an
envelope. For instance, the condition that the line of coordinates (l,
m, n), i.e. the line of which the allied point-coordinate equation is
lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)^2 =
0, is readily found to be of the form (A, B, C, F, G, H) (l, m, n)^2 =
0, i.e. to be of the second degree in the line-coordinates. It is not
hard to show that the _general_ equation of the second degree in l, m,
n thus represents a conic; but the degenerate conics of
line-coordinates are not line-pairs, as in point-coordinates, but
point-pairs.
The degree of the point-coordinate equation of a curve is the _order_
of the curve, the number of points in which it cuts a straight line.
That of the line-coordinate equation is its _class_, the number of
tangents to it from a point. The order and class of a curve are
generally different when either exceeds two.
26. The system of line-coordinates allied to the areal system of
point-coordinates has special interest.
The l, m, n of this system are the perpendiculars [xi], [eta], [zeta]
of S 24; and x'[xi] + y'[eta] + z'[zeta] = 0 is the equation of the
point of areal coordinates (x', y', z'), i.e. is a relation which the
perpendiculars from the vertices of the triangle of reference on every
line through the point, but no other line, satisfy. Notice that a
non-homogeneous equation of the first degree in [xi], [eta], [zeta]
does not, as a homogeneous one does, represent a point, but a circle.
In fact x'[xi] + y'[eta] + z'[zeta] = R expresses the constancy of the
perpendicular distance of the fixed point x'[xi] + y'[eta] + z'[zeta]
= 0 from the variable line ([xi], [eta], [zeta]), i.e. the fact that
([xi], [eta], [zeta]) touches a circle with the fixed point for
centre. The relation in any [xi], [eta], [zeta] which enables us to
make an equation homogeneous is not linear, as in point-coordinates,
but quadratic, viz. it is the relation {a[xi], b[eta], c[zeta]}^2 =
4[Delta]^2 of S 24. Accordingly the homogeneous equation of the above
circle is
4[Delta]^2(x'[xi] + y'[eta] + z'[zeta])^2 = R^2{a[xi], b[eta], c[zeta]}^2.
Every circle has an equation of this form in the present system of
line-coordinates. Notice that the equation of any circle is satisfied
by those coordinates of lines which satisfy both x'[xi] + y'[eta] +
z'[zeta] = 0, the equation of its centre, and {a[xi], b[eta],
c[zeta]}^2 = 0. This last equation, of which the left-hand side
satisfies the condition for breaking up into two factors, represents
the two imaginary circular points at infinity, through which all
circles and their asymptotes pass.
There is strict duality in descriptive geometry between
point-line-locus and line-point-envelope theorems. But in metrical
geometry duality is encumbered by the fact that there is in a plane
one special line only, associated with distance, while of special
points, associated with direction, there are two: moreover the line is
real, and the points both imaginary.
II. _Solid Analytical Geometry._
27. Any point in space may be specified by three coordinates. We
consider three fixed planes of reference, and generally, as in all
that follows, three which are at right angles two and two. They
intersect, two and two, in lines x'Ox, y'Oy, z'Oz, called the axes of
x, y, z respectively, and divide all space into eight parts called
octants. If from any point P in space we draw PN parallel to zOz' to
meet the plane xOy in N, and then from N draw NM parallel to yOy' to
meet x'Ox in M, the coordinates (x, y, z) of P are the numerical
measures of OM, MN, NP; in the case of rectangular coordinates these
are the perpendicular distances of P from the three planes of
reference. The sign of each coordinate is positive or negative as P
lies on one side or the other of the corresponding plane. In the
octant delineated the signs are taken all positive.
[Illustration: FIG. 57.]
[Illustration: FIG. 58.]
In fig. 57 the delineation is on a plane of the paper taken parallel
to the plane zOx, the points of a solid figure being projected on that
plane by parallels to some chosen line through O in the positive
octant. Sometimes it is clearer to delineate, as in fig. 58, by
projection parallel to that line in the octant which is equally
inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it.
It is possible by parallel projection to delineate equal scales along
Ox, Oy, Oz by scales having any ratios we like along lines in a plane
having any mutual inclinations we like.
[Illustration: FIG. 59.]
For the delineation of a surface of simple form it frequently suffices
to delineate the sections by the coordinate planes; and, in
particular, when the surface has symmetry about each coordinate plane,
to delineate the quarter-sections belonging to a single octant. Thus
fig. 59 conveniently represents an octant of the wave surface, which
cuts each coordinate plane in a circle and an ellipse. Or we may
delineate a series of contour lines, i.e. sections by planes parallel
to xOy, or some other chosen plane; of course other sections may be
indicated too for greater clearness. For the delineation of a curve a
good method is to represent, as above, a series of points P thereof,
each accompanied by its ordinate PN, which serves to refer it to the
plane of xy. The employment of stereographic projection is also
interesting.
28. In plane geometry, reckoning the line as a curve of the first
order, we have only the point and the curve. In solid geometry,
reckoning a line as a curve of the first order, and the plane as a
surface of the first order, we have the point, the curve and the
surface; but the increase of complexity is far greater than would
hence at first sight appear. In plane geometry a curve is considered
in connexion with lines (its tangents); but in solid geometry the
curve is considered in connexion with lines and planes (its tangents
and osculating planes), and the surface also in connexion with lines
and planes (its tangent lines and tangent planes); there are surfaces
arising out of the line--cones, skew surfaces, developables, doubly
and triply infinite systems of lines, and whole classes of theories
which have nothing analogous to them in plane geometry: it is thus a
very small part indeed of the subject which can be even referred to in
the present article.
In the case of a surface we have between the coordinates (x, y, z) a
single, or say a onefold relation, which can be represented by a
single relation [f](x, y, z) = 0; or we may consider the coordinates
expressed each of them as a given function of two variable parameters
p, q; the form z = [f](x, y) is a particular case of each of these
modes of representation; in other words, we have in the first mode
[f](x, y, z) = z - [f](x, y), and in the second mode x = p, y = q for
the expression of two of the coordinates in terms of the parameters.
In the case of a curve we have between the coordinates (x, y, z) a
twofold relation: two equations [f](x, y, z) = 0, [phi](x, y, z) = 0
give such a relation; i.e. the curve is here considered as the
intersection of two surfaces (but the curve is not always the complete
intersection of two surfaces, and there are hence difficulties); or,
again, the coordinates may be given each of them as a function of a
single variable parameter. The form y = [phi](x), z = [psi](x), where
two of the coordinates are given in terms of the third, is a
particular case of each of these modes of representation.
29. The remarks under plane geometry as to descriptive and metrical
propositions, and as to the non-metrical character of the method of
coordinates when used for the proof of a descriptive proposition,
apply also to solid geometry; and they might be illustrated in like
manner by the instance of the theorem of the radical centre of four
spheres. The proof is obtained from the consideration that S and S'
being each of them a function of the form x^2 + y^2 + z^2 + ax + by +
cz + d, the difference S-S' is a mere linear function of the
coordinates, and consequently that S-S' = 0 is the equation of the
plane containing the circle of intersection of the two spheres S = 0
and S' = 0.
[Illustration: FIG. 60.]
30. _Metrical Theory._--The foundation in solid geometry of the
metrical theory is in fact the before-mentioned theorem that if a
finite right line PQ be projected upon any other line OO' by lines
perpendicular to OO', then the length of the projection P'Q' is equal
to the length of PQ into the cosine of its inclination to P'Q'--or (in
the form in which it is now convenient to state the theorem) the
perpendicular distance P'Q' of two parallel planes is equal to the
inclined distance PQ into the cosine of the inclination. The principle
of S 16, that the algebraical sum of the projections of the sides of
any closed polygon on any line is zero, or that the two sets of sides
of the polygon which connect a vertex A and a vertex B have the same
sum of projections on the line, in sign and magnitude, as we pass from
A to B, is applicable when the sides do not all lie in one plane.
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being
respectively parallel to the three rectangular axes Ox, Oy, Oz; let
the lengths of these sides be [xi], [eta], [zeta], and that of the
side QP be = [rho]; and let the cosines of the inclinations (or say
the cosine-inclinations) of [rho] to the three axes be [alpha],
[beta], [gamma]; then projecting successively on the three sides and
on QP we have
[xi], [eta], [zeta] = [rho][alpha], [rho][beta], [rho][gamma],
and
[rho] = [alpha][xi] + [beta][eta] + [gamma][zeta],
whence [rho]^2 = [xi]^2 + [eta]^2 + [zeta]^2, which is the relation
between a distance [rho] and its projections [xi], [eta], [zeta] upon
three rectangular axes. And from the same equations we obtain
[alpha]^2 + [beta]^2 + [gamma]^2 = 1, which is a relation connecting
the cosine-inclinations of a line to three rectangular axes.
Suppose we have through Q any other line QT, and let the
cosine-inclinations of this to the axes be [alpha]', [beta]',
[gamma]', and [delta] be its cosine-inclination to QP; also let [rho]
be the length of the projection of QP upon QT; then projecting on QT
we have
[rho] = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta] = [rho][delta].
And in the last equation substituting for [xi], [eta], [zeta] their
values [rho][alpha], [rho][beta], [rho][gamma] we find
[delta] = [alpha][alpha]' + [beta][beta]' + [gamma][gamma]',
which is an expression for the mutual cosine-inclination of two lines,
the cosine-inclinations of which to the axes are [alpha], [beta],
[gamma] and [alpha]', [beta]', [gamma]' respectively. We have of
course [alpha]^2 + [beta]^2 + [gamma]^2 = 1 and [alpha]'^2 + [beta]'^2
+ [gamma]'^2 = 1; and hence also
1 - [delta]^2 = ([alpha]^2 + [beta]^2 + [gamma]^2)([alpha]'^2 + [beta]'^2 + [gamma]'^2)
- ([alpha][alpha]' + [beta][beta]' + [gamma][gamma]')^2,
= ([beta][gamma]' - [beta]'[gamma])^2 + ([gamma][alpha]' - [gamma]'[alpha])^2 +
([alpha][beta]' - [alpha]'[beta])^2;
so that the sine of the inclination can only be expressed as a square
root. These formulae are the foundation of spherical trigonometry.
32. _Straight Lines, Planes and Spheres._--The foregoing formulae give
at once the equations of these loci.
For first, taking Q to be a fixed point, coordinates (a, b, c), and
the cosine-inclinations ([alpha], [beta], [gamma]) to be constant,
then P will be a point in the line through Q in the direction thus
determined; or, taking (x, y, z) for its coordinates, these will be
the current coordinates of a point in the line. The values of [xi],
[eta], [zeta] then are x - a, y - b, z - c, and we thus have
x - a y - b z - c
------- = ----- = ------- (= [rho]),
[alpha] [beta] [gamma]
which (omitting the last equation, = [rho]) are the equations of the
line through the point (a, b, c), the cosine-inclinations to the axes
being [alpha], [beta], [gamma], and these quantities being connected
by the relation [alpha]^2 + [beta]^2 + [gamma]^2 = 1. This equation
may be omitted, and then [alpha], [beta], [gamma], instead of being
equal, will only be proportional, to the cosine-inclinations.
Using the last equation, and writing
x, y, z = a + [alpha][rho], b + [beta][rho], c + [gamma][rho],
these are expressions for the current coordinates in terms of a
parameter [rho], which is in fact the distance from the fixed point
(a, b, c).
It is easy to see that, if the coordinates (x, y, z) are connected by
any two linear equations, these equations can always be brought into
the foregoing form, and hence that the two linear equations represent
a line.
Secondly, taking for greater simplicity the point Q to be coincident
with the origin, and [alpha]', [beta]', [gamma]', p to be constant,
then p is the perpendicular distance of a plane from the origin, and
[alpha]', [beta]', [gamma]' are the cosine-inclinations of this
distance to the axes ([alpha]'^2 + [beta]'^2 + [gamma]'^2 = 1). P is
any point in this plane, and taking its coordinates to be (x, y, z)
then ([xi], [eta], [zeta]) are = (x, y, z), and the foregoing equation
p = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta] becomes
[alpha]'x + [beta]'y + [gamma]'z = p,
which is the equation of the plane in question.
If, more generally, Q is not coincident with the origin, then, taking
its coordinates to be (a, b, c), and writing p1 instead of p, the
equation is
[alpha]'(x - a) + [beta]'(y - b) + [gamma]'(z - c) = p1;
and we thence have p1 = p - (a[alpha]' + b[beta]' + c[gamma]'), which
is an expression for the perpendicular distance of the point (a, b, c)
from the plane in question.
It is obvious that any linear equation Ax + By + Cz + D = O between
the coordinates can always be brought into the foregoing form, and
hence that such an equation represents a plane.
Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and
the distance QP = [rho], to be constant, say this is = d, then, as
before, the values of [xi], [eta], [zeta] are x - a, y - b, z - c, and
the equation [xi]^2 + [eta]^2 + [zeta]^2 = [rho]^2 becomes
(x - a)^2 + (y - b)^2 + (z - c)^2 = d^2,
which is the equation of the sphere, coordinates of the centre = (a,
b, c), and radius = d.
A quadric equation wherein the terms of the second order are x^2 + y^2
+ z^2, viz. an equation
x^2 + y^2 + z^2 + Ax + By + Cz + D = 0,
can always, it is clear, be brought into the foregoing form; and it
thus appears that this is the equation of a sphere, coordinates of the
centre -1/2A, -1/2B, -1/2C, and squared radius = 1/4(A^2 + B^2 + C^2)
- D.
33. _Cylinders, Cones, ruled Surfaces._--If the two equations of a
straight line involve a parameter to which any value may be given, we
have a singly infinite system of lines. They cover a surface, and the
equation of the surface is obtained by eliminating the parameter
between the two equations.
If the lines all pass through a given point, then the surface is a
cone; and, in particular, if the lines are all parallel to a given
line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the
line x = mz, y = nz, the equations of a line of the system are x = mz
+ a, y = nz + b,--where a, b are supposed to be functions of the
variable parameter, or, what is the same thing, there is between them
a relation f(a, b) = 0: we have a = x - mz, b = y - nz, and the result
of the elimination of the parameter therefore is [f](x - mz, y - nz) =
0, which is thus the general equation of the cylinder the generating
lines whereof are parallel to the line x = mz, y = nz. The equation of
the section by the plane z = 0 is [f](x, y) = 0, and conversely if the
cylinder be determined by means of its curve of intersection with the
plane z = 0, then, taking the equation of this curve to be f(x, y) =
0, the equation of the cylinder is [f](x - mz, y - nz) = 0. Thus, if
the curve of intersection be the circle (x - [alpha])^2 + (y -
[beta])^2 = [gamma]^2, we have (x - mz - [alpha])^2 + (y - nz -
[beta])^2 = [gamma]^2 as the equation of an oblique cylinder on this
base, and thus also (x - [alpha])^2 + (y - [beta])^2 = [gamma]^2 as
the equation of the right cylinder.
If the lines all pass through a given point (a, b, c), then the
equations of a line are x - a = [alpha](z - c), y - b = [beta](z - c),
where [alpha], [beta] are functions of the variable parameter, or,
what is the same thing, there exists between them an equation
f([alpha], [beta]) = 0; the elimination of the parameter gives,
therefore, f[(x - a)/(x - c'), (y - b)/(z - c)] = 0; and this
equation, or, what is the same thing, any homogeneous equation f(x -
a, y - b, z - c) = 0, or, taking f to be a rational and integral
function of the order n, say (*)(x - a, y - b, z - c)^n = 0, is the
general equation of the cone having the point (a, b, c) for its
vertex. Taking the vertex to be at the origin, the equation is (*)(x,
y, z)^n = 0; and, in particular, (*)(x, y, z)^2 = 0 is the equation of
a cone of the second order, or quadricone, having the origin for its
vertex.
34. In the general case of a singly infinite system of lines, the
locus is a ruled surface (or _regulus_). Now, when a line is changing
its position in space, it may be looked upon as in a state of turning
about some point in itself, while that point is, as a rule, in a state
of moving out of the plane in which the turning takes place. If
instantaneously it is only in a state of turning, it is usual, though
not strictly accurate, to say that it intersects its consecutive
position. A regulus such that consecutive lines on it do not
intersect, in this sense, is called a skew surface, or _scroll_; one
on which they do is called a developable surface or _torse_.
Suppose, for instance, that the equations of a line (depending on the
variable parameter [theta]) are x/a + y/c = [theta] (1 + y/b), x/a -
z/c = 1/[theta] (1 - y/b); then, eliminating [theta] we have x^2/a^2 -
z^2/c^2 = 1 - y^2/b^2, or say, x^2/a^2 + z^2/b^2 - z^2/c^2 = 1, the
equation of a quadric surface, afterwards called the hyperboloid of
one sheet; this surface is consequently a scroll. It is to be remarked
that we have upon the surface a second singly infinite series of
lines; the equations of a line of this second system (depending on the
variable parameter [phi]) are
x z / y \ x z 1 / y \
-- + -- = [phi]( 1 - -- ), -- - -- = ----- ( 1 + -- ).
a c \ b / a c [phi] \ b /
It is easily shown that any line of the one system intersects every
line of the other system.
Considering any curve (of double curvature) whatever, the tangent
lines of the curve form a singly infinite system of lines, each line
intersecting the consecutive line of the system,--that is, they form a
developable, or torse; the curve and torse are thus inseparably
connected together, forming a single geometrical figure. An osculating
plane of the curve (see S 38 below) is a tangent plane of the torse
all along a generating line.
35. _Transformation of Coordinates._--There is no difficulty in
changing the origin, and it is for brevity assumed that the origin
remains unaltered. We have, then, two sets of rectangular axes, Ox,
Oy, Oz, and Ox1, Oy1, Ozx1, the mutual cosine-inclinations being shown
by the diagram--
| x | y | z |
----+---------+--------+---------+
x1 | [alpha] | [beta] | [gamma] |
----+---------+--------+---------+
y1 | [alpha] | [beta]'| [gamma]'|
----+---------+--------+---------+
z1 | [alpha]"| [beta]"| [gamma]"|
----+---------+--------+---------+
that is, [alpha], [beta], [gamma] are the cosine-inclinations of Ox1
to Ox, Oy, Oz; [alpha]', [beta]', [gamma]' those of Oy1, &c.
And this diagram gives also the linear expressions of the coordinates
(x1, y1, z1) or (x, y, z) of either set in terms of those of the other
set; we thus have
x1 = [alpha] x + [beta] y + [gamma] z,
x = [alpha]x1 + [alpha]'y1 + [alpha]"z1,
y1 = [alpha]'x + [beta]'y + [gamma]'z,
y = [beta]x1 + [beta]'y1 + [beta]"z1,
z1 = [alpha]"x + [beta]"y + [gamma]"z,
z = [gamma]x1 + [gamma]'y1 + [gamma]"z1,
which are obtained by projection, as above explained. Each of these
equations is, in fact, nothing else than the before-mentioned equation
p = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta], adapted to the
problem in hand.
But we have to consider the relations between the nine coefficients.
By what precedes, or by the consideration that we must have
identically x^2 + y^2 + z^2 = x1^2 + y1^2 + z1^2, it appears that
these satisfy the relations--
a^2 + [beta]^2 + [gamma]^2 = 1,
[alpha]^2 + [alpha]'^2 + [alpha]"^2 = 1,
[alpha]'^2 + [beta]'^2 + [gamma]'^2 = 1,
[beta]^2 + [beta]'^2 + [beta]"^2 = 1,
[alpha]"^2 + [beta]"^2 + [gamma]"^2 = 1,
[gamma]^2 + [gamma]'^2 + [gamma]"^2 = 1,
a'a" + [beta]'[beta]" + [gamma]'[gamma]" = 0,
[beta][gamma] +[beta]'[gamma]' + [beta]"[gamma]" = 0,
[alpha]"[alpha] + [beta]"[beta] + [gamma]"[gamma] = 0,
[gamma][alpha] + [gamma]'[alpha]' + [gamma]"[alpha]" = 0,
[alpha][alpha]' + [beta][beta]' + [gamma][gamma]' = 0,
[alpha][beta] +[alpha]'[beta]' + [alpha]"[beta]" = 0,
either set of six equations being implied in the other set.
It follows that the square of the determinant
|[alpha], [beta], [gamma] |
| |
|[alpha]', [beta]', [gamma]'|
| |
|[alpha]", [beta]", [gamma]"|
is = 1; and hence that the determinant itself is = [+-] 1. The
distinction of the two cases is an important one: if the determinant
is = + 1, then the axes Ox1, Oy1, Oz1 are such that they can by a
rotation about O be brought to coincide with Ox, Oy, Oz respectively;
if it is = -1, then they cannot. But in the latter case, by measuring
x1, y1, z1 in the opposite directions we change the signs of all the
coefficients and so make the determinant to be = + 1; hence the former
case need alone be considered, and it is accordingly assumed that the
determinant is = + 1. This being so, it is found that we have the
equality [alpha] = [beta]'[gamma]" - [beta]"[gamma]', and eight like
ones, obtained from this by cyclical interchanges of the letters
[alpha], [beta], [gamma], and of unaccented, singly and doubly
accented letters.
36. The nine cosine-inclinations above are, as has been seen,
connected by six equations. It ought then to be possible to express
them all in terms of three parameters. An elegant means of doing this
has been given by Rodrigues, who has shown that the tabular expression
of the formulae of transformation may be written
| x | y | z |
----+--------------------------------+--------------------------------+--------------------------------+
x1 |1 + [lambda]^2 - [mu]^2 - [nu]^2| 2([lambda][mu] - [nu]) | 2([nu][lambda] + [mu]) |
----+--------------------------------+--------------------------+-----+--------------------------------+
y1 | 2([lambda][mu] + [nu]) |1 - [lambda]^2 + [mu]^2 - [nu]^2| 2([mu][nu] + [lambda]) |
----+--------------------------------+--------------------------+-----+--------------------------------+
z1 | 2([nu][lambda] - [mu]) | 2([mu][nu] + [lambda]) |1 - [lambda]^2 - [mu]^2 + [nu]^2|
----+--------------------------------+--------------------------------+--------------------------------+
/(1 + [lambda]^2 + [mu]^2 + [nu]^2),
the meaning being that the coefficients in the transformation are
fractions, with numerators expressed as in the table, and the common
denominator.
37. _The Species of Quadric Surfaces_.--Surfaces represented by
equations of the second degree are called _quadric_ surfaces. Quadric
surfaces are either _proper_ or _special_. The special ones arise when
the coefficients in the general equation are limited to satisfy
certain special equations; they comprise (1) plane-pairs, including in
particular one plane twice repeated, and (2) cones, including in
particular cylinders; there is but one form of cone, but cylinders may
be elliptic, parabolic or hyperbolic.
A discussion of the general equation of the second degree shows that
the _proper_ quadric surfaces are of five kinds, represented
respectively, when referred to the most convenient axes of reference,
by equations of the five types (a and b positive):
x^2 y^2
(1) z = --- + ---, elliptic paraboloid.
2a 2b
x^2 y^2
(2) z = --- - ---, hyperbolic paraboloid.
2a 2b
x^2 y^2 z^2
(3) --- + --- + --- = 1, ellipsoid.
a^2 b^2 c^2
x^2 y^2 z^2
(4) --- + --- - --- = 1, hyperboloid of one sheet.
a^2 b^2 c^2
x^2 y^2 z^2
(5) --- + --- - --- = -1, hyperboloid of two sheets.
a^2 b^2 c^2
It is at once seen that these are distinct surfaces; and the equations
also show very readily the general form and mode of generation of the
several surfaces.
[Illustration: FIG. 61.]
In the elliptic paraboloid (fig. 61) the sections by the planes of zx
and zy are the parabolas
x^2 y^2
z = ---, z = ---
2a 2b
having the common axes Oz; and the section by any plane z = [gamma]
parallel to that of xy is the ellipse
x^2 y^2
[gamma] = --- + ---;
2a 2b
so that the surface is generated by a variable ellipse moving parallel
to itself along the parabolas as directrices.
[Illustration: FIG. 62.]
[Illustration: FIG. 63.]
In the hyperbolic paraboloid (figs. 62 and 63) the sections by the
planes of zx, zy are the parabolas z = x^2/2a, z = - y^2/2b, having
the opposite axes Oz, Oz', and the section by a plane z = [gamma]
parallel to that of xy is the hyperbola [gamma] = x^2/2a - y^2/2b,
which has its transverse axis parallel to Ox or Oy according as
[gamma] is positive or negative. The surface is thus generated by a
variable hyperbola moving parallel to itself along the parabolas as
directrices. The form is best seen from fig. 63, which represents the
sections by planes parallel to the plane of xy, or say the contour
lines; the continuous lines are the sections above the plane of xy,
and the dotted lines the sections below this plane. The form is, in
fact, that of a saddle.
[Illustration: FIG. 64.]
In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and
xy are each of them an ellipse, and the section by any parallel plane
is also an ellipse. The surface may be considered as generated by an
ellipse moving parallel to itself along two ellipses as directrices.
In the hyperboloid of one sheet (fig. 65), the sections by the planes
of zx, zy are the hyperbolas
x^2 z^2 y^2 z^2
--- - --- = 1, --- - --- = 1,
c^2 c^2 b^2 c^2
having a common conjugate axis zOz'; the section by the plane of x, y,
and that by any parallel plane, is an ellipse; and the surface may be
considered as generated by a variable ellipse moving parallel to
itself along the two hyperbolas as directrices. If we imagine two
equal and parallel circular disks, their points connected by strings
of equal lengths, so that these are the generators of a right circular
cylinder, and if we turn one of the disks about its centre through an
angle in its plane, the strings in their new positions will be one
system of generators of a hyperboloid of one sheet, for which a = b;
and if we turn it through the same angle in the opposite direction, we
get in like manner the generators of the other system; there will be
the same general configuration when a = | b. The hyperbolic paraboloid
is also covered by two systems of rectilinear generators as a method
like that used in S 34 establishes without difficulty. The figures
should be studied to see how they can lie.
[Illustration: FIG. 65.]
[Illustration: FIG. 66.]
In the hyperboloid of two sheets (fig. 66) the sections by the planes
of zx and zy are the hyperbolas
z^2 x^2 z^2 y^2
--- - --- = 1, --- - --- = 1,
c^2 a^2 c^2 b^2
having a common transverse axis along z'Oz; the section by any plane z
= [+-][gamma] parallel to that of xy is the ellipse
x^2 y^2 [gamma]^2
--- + --- = --------- - 1,
a^2 b^2 c^2
provided [gamma]^2 > c^2, and the surface, consisting of two distinct
portions or sheets, may be considered as generated by a variable
ellipse moving parallel to itself along the hyperbolas as directrices.
38. _Differential Geometry of Curves._--For convenience consider the
coordinates (x, y, z) of a point on a curve in space to be given as
functions of a variable parameter [theta], which may in particular be
one of themselves. Use the notation x', x" for dx/d[theta],
d^2x/d[theta]^2, and similarly as to y and z. Only a few formulae will
be given. Call the current coordinates ([xi], [eta], [zeta]).
The _tangent_ at (x, y, z) is the line tended to as a limit by the
connector of (x, y, z) and a neighbouring point of the curve when the
latter moves up to the former: its equations are
([xi] - x)/x' = ([eta] - y)/y' = ([zeta] - z)/z'.
The _osculating plane_ at (x, y, z) is the plane tended to as a limit
by that through (x, y, z) and two neighbouring points of the curve as
these, remaining distinct, both move up to (x, y, z): its one equation
is
([xi] - x)(y'z" - y"z') + ([eta] - y)(z'x" - z"x') + ([zeta] - z)
(x'y" - x"y') = 0.
The _normal plane_ is the plane through (x, y, z) at right angles to
the tangent line, i.e. the plane
x'([xi] - x) + y'([eta] - y) + z'([zeta] - z) = 0.
It cuts the osculating plane in a line called the _principal normal_.
Every line through (x, y, z) in the normal plane is a normal. The
normal perpendicular to the osculating plane is called the _binormal_.
A tangent, principal normal, and binormal are a convenient set of
rectangular axes to use as those of reference, when the nature of a
curve near a point on it is to be discussed.
Through (x, y, z) and three neighbouring points, all on the curve,
passes a single sphere; and as the three points all move up to (x, y,
z) continuing distinct, the sphere tends to a limiting size and
position. The limit tended to is the sphere of closest contact with
the curve at (x, y, z); its centre and radius are called the centre
and radius of _spherical curvature_. It cuts the osculating plane in a
circle, called the _circle of absolute curvature_; and the centre and
radius of this circle are the centre and radius of absolute curvature.
The centre of absolute curvature is the limiting position of the point
where the principal normal at (x, y, z) is cut by the normal plane at
a neighbouring point, as that point moves up to (x, y, z).
39. _Differential Geometry of Surfaces._--Let (x, y, z) be any chosen
point on a surface [f](x, y, z) = 0. As a second point of the surface
moves up to (x, y, z), its connector with (x, y, z) tends to a
limiting position, a tangent line to the surface at (x, y, z). All
these tangent lines at (x, y, z), obtained by approaching (x, y, z)
from different directions on a surface, lie in one plane
dP[f] dP[f] dP[f]
----- ([xi] - x) + ----- ([eta] - y) + ----- ([zeta] - z) = 0.
dPx dPy dPz
This plane is called the _tangent plane_ at (x, y, z). One line
through (x, y, z) is at right angles to the tangent plane. This is the
normal
/dP[f] /dP[f] /dP[f]
([xi] - x) / ----- = ([eta] - y) / ----- = ([zeta] - z) = / -----.
/ dPx / dPy / dPz
The tangent plane is cut by the surface in a curve, real or imaginary,
with a node or double point at (x, y, z). Two of the tangent lines
touch this curve at the node. They are called the "chief tangents"
(_Haupt-tangenten_) at (x, y, z); they have closer contact with the
surface than any other tangents.
In the case of a quadric surface the curve of intersection of a
tangent and the surface is of the second order and has a node, it must
therefore consist of two straight lines. Consequently a quadric
surface is covered by two sets of straight lines, a pair through every
point on it; these are imaginary for the ellipsoid, hyperboloid of two
sheets, and elliptic paraboloid.
A surface of any order is covered by two singly infinite systems of
curves, a pair through every point, the tangents to which are all
chief tangents at their respective points of contact. These are called
_chief-tangent curves_; on a quadric surface they are the above
straight lines.
40. The tangents at a point of a surface which bisect the angles
between the chief tangents are called the _principal tangents_ at the
point. They are at right angles, and together with the normal
constitute a convenient set of rectangular axes to which to refer the
surface when its properties near the point are under discussion. At a
special point which is such that the chief tangents there run to the
circular points at infinity in the tangent plane, the principal
tangents are indeterminate; such a special point is called an umbilic
of the surface.
There are two singly infinite systems of curves on a surface, a pair
cutting one another at right angles through every point upon it, all
tangents to which are principal tangents of the surface at their
respective points of contact. These are called _lines of curvature_,
because of a property next to be mentioned.
As a point Q moves in an arbitrary direction on a surface from
coincidence with a chosen point P, the normal at it, as a rule, at
once fails to meet the normal at P; but, if it takes the direction of
a line of curvature through P, this is instantaneously not the case.
We have thus on the normal two centres of curvature, and the distances
of these from the point on the surface are the two _principal radii of
curvature_ of the surface at that point; these are also the radii of
curvature of the sections of the surface by planes through the normal
and the two principal tangents respectively; or say they are the radii
of curvature of the normal sections through the two principal tangents
respectively. Take at the point the axis of z in the direction of the
normal, and those of x and y in the directions of the principal
tangents respectively, then, if the radii of curvature be a, b (the
signs being such that the coordinates of the two centres of curvature
are z = a and z = b respectively), the surface has in the
neighbourhood of the point the form of the paraboloid
x^2 y^2
z = --- + ---,
2a 2b
and the chief-tangents are determined by the equation 0 = x^2/2a +
y^2/2b. The two centres of curvature may be on the same side of the
point or on opposite sides; in the former case a and b have the same
sign, the paraboloid is elliptic, and the chief-tangents are
imaginary; in the latter case a and b have opposite signs, the
paraboloid is hyperbolic, and the chief-tangents are real.
The normal sections of the surface and the paraboloid by the same
plane have the same radius of curvature; and it thence readily follows
that the radius of curvature of a normal section of the surface by a
plane inclined at an angle [theta] to that of zx is given by the
equation
1 cos^2 [theta] sin^2 [theta]
----- = ------------- + -------------.
[rho] a b
The section in question is that by a plane through the normal and a
line in the tangent plane inclined at an angle [theta] to the
principal tangent along the axis of x. To complete the theory,
consider the section by a plane having the same trace upon the tangent
plane, but inclined to the normal at an angle [phi]; then it is shown
without difficulty (Meunier's theorem) that the radius of curvature of
this inclined section of the surface is = [rho] cos [phi].
AUTHORITIES.--The above article is largely based on that by Arthur
Cayley in the 9th edition of this work. Of early and important recent
publications on analytical geometry, special mention is to be made of
R. Descartes, _Geometrie_ (Leyden, 1637); John Wallis, _Tractatus de
sectionibus conicis nova methodo expositis_ (1655, _Opera
mathematica_, i., Oxford, 1695); de l'Hospital, _Traite analytique des
sections coniques_ (Paris, 1720); Leonhard Euler, _Introductio in
analysin infinitorum_, ii. (Lausanne, 1748); Gaspard Monge,
"Application d'algebre a la geometrie" (_Journ. Ecole Polytech._,
1801); Julius Plucker, _Analytisch-geometrische Entwickelungen_, 3
Bde. (Essen, 1828-1831); _System der analytischen Geometrie_ (Berlin,
1835); G. Salmon, _A Treatise on Conic Sections_ (Dublin, 1848; 6th
ed., London, 1879); Ch. Briot and J. Bouquet, _Lecons de geometrie
analytique_ (Paris, 1851; 16th ed., 1897); M. Chasles, _Traite de
geometrie superieure_ (Paris, 1852); Wilhelm Fiedler, _Analytische
Geometrie der Kegelschnitte_ nach G. Salmon frei bearbeitet (Leipzig,
5te Aufl., 1887-1888); N.M. Ferrers, _An Elementary Treatise on
Trilinear Coordinates_ (London, 1861); Otto Hesse, _Vorlesungen aus
der analytischen Geometrie_ (Leipzig, 1865, 1881); W.A. Whitworth,
_Trilinear Coordinates and other Methods of Modern Analytical
Geometry_ (Cambridge, 1866); J. Booth, _A Treatise on Some New
Geometrical Methods_ (London, i., 1873; ii., 1877); A. Clebsch-F.
Lindemann, _Vorlesungen uber Geometrie_, Bd. i. (Leipzig, 1876, 2te
Aufl., 1891); R. Baltser, _Analytische Geometrie_ (Leipzig, 1882);
Charlotte A. Scott, _Modern Methods of Analytical Geometry_ (London,
1894); G. Salmon, _A Treatise on the Analytical Geometry of three
Dimensions_ (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler,
_Analytische Geometrie des Raumes_ (Leipzig, 1863; 4te Aufl., 1898);
P. Frost, _Solid Geometry_ (London, 3rd ed., 1886; 1st ed., Frost and
J. Wolstenholme). See also E. Pascal, _Repertorio di matematiche
superiori, II. Geometria_ (Milan, 1900), and articles now appearing in
the _Encyklopadie der mathematischen Wissenschaften_, Bd. iii. 1, 2.
(E. B. El.)
V. LINE GEOMETRY
Line geometry is the name applied to those geometrical investigations in
which the straight line replaces the point as element. Just as ordinary
geometry deals primarily with points and systems of points, this theory
deals in the first instance with straight lines and systems of straight
lines. In two dimensions there is no necessity for a special line
geometry, inasmuch as the straight line and the point are
interchangeable by the principle of duality; but in three dimensions the
straight line is its own reciprocal, and for the better discussion of
systems of lines we require some new apparatus, e.g., a system of
coordinates applicable to straight lines rather than to points. The
essential features of the subject are most easily elucidated by
analytical methods: we shall therefore begin with the notion of line
coordinates, and in order to emphasize the merits of the system of
coordinates ultimately adopted, we first notice a system without these
advantages, but often useful in special investigations.
In ordinary Cartesian coordinates the two equations of a straight line
may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u may
be regarded as the four coordinates of the line. These coordinates
lack symmetry: moreover, in changing from one base of reference to
another the transformation is not linear, so that the degree of an
equation is deprived of real significance. For purposes of the general
theory we employ homogeneous coordinates; if x1y1z1w1 and x2y2z2w2 are
two points on the line, it is easily verified that the six
determinants of the array
|x1y1z1w1|
|x2y2z2w2|
are in the same ratios for all point-pairs on the line, and further,
that when the point coordinates undergo a linear transformation so
also do these six determinants. We therefore adopt these six
determinants for the coordinates of the line, and express them by the
symbols l, [lambda], m, [mu], n, [nu] where l = x1w2 - x2w1, [lambda]
= y1z2 - y2z1, &c. There is the further advantage that if a1b1c1d1 and
a2b2c2d2 be two planes through the line, the six determinants
|a1b1c1d1|
|a2b2c2d2|
are in the same ratios as the foregoing, so that except as regards a
factor of proportionality we have [lambda] = b1c2 - b2c1, l = c1d2 -
c2d1, &c. The identical relation l[lambda] + m[mu] + n[nu] = o reduces
the number of independent constants in the six coordinates to four,
for we are only concerned with their mutual ratios; and the quadratic
character of this relation marks an essential difference between point
geometry and line geometry. The condition of intersection of two lines
is
l[lambda]' + l'[lambda] + m[mu]' + m'[mu] + n[nu]' + n'[nu] = 0
where the accented letters refer to the second line. If the
coordinates are Cartesian and l, m, n are direction cosines, the
quantity on the left is the mutual moment of the two lines.
Since a line depends on four constants, there are three distinct types
of configurations arising in line geometry--those containing a
triply-infinite, a doubly-infinite and a singly-infinite number of
lines; they are called Complexes, Congruences, and Ruled Surfaces or
Skews respectively. A _Complex_ is thus a system of lines satisfying
one condition--that is, the coordinates are connected by a single
relation; and the degree of the complex is the degree of this equation
supposing it to be algebraic. The lines of a complex of the nth degree
which pass through any point lie on a cone of the nth degree, those
which lie in any plane envelop a curve of the nth class and there are
n lines of the complex in any plane pencil; the last statement
combines the former two, for it shows that the cone is of the nth
degree and the curve is of the nth class. To find the lines common to
four complexes of degrees n1, n2, n3, n4, we have to solve five
equations, viz. the four complex equations together with the quadratic
equation connecting the line coordinates, therefore the number of
common lines is 2n1n2n3n4. As an example of complexes we have the
lines meeting a twisted curve of the nth degree, which form a complex
of the nth degree.
A _Congruence_ is the set of lines satisfying two conditions: thus a
finite number m of the lines pass through any point, and a finite
number n lie in any plane; these numbers are called the degree and
class respectively, and the congruence is symbolically written (m, n).
The simplest example of a congruence is the system of lines
constituted by all those that pass through m points and those that lie
in n planes; through any other point there pass m of these lines, and
in any other plane there lie n, therefore the congruence is of degree
m and class n. It has been shown by G.H. Halphen that the number of
lines common to two congruences is mm' + nn', which may be verified by
taking one of them to be of this simple type. The lines meeting two
fixed lines form the general (1, 1) congruence; and the chords of a
twisted cubic form the general type of a (1, 3) congruence; Halphen's
result shows that two twisted cubics have in general ten common
chords. As regards the analytical treatment, the difficulty is of the
same nature as that arising in the theory of curves in space, for a
congruence is not in general the complete intersection of two
complexes.
A _Ruled Surface_, _Regulus_ or _Skew_ is a configuration of lines
which satisfy three conditions, and therefore depend on only one
parameter. Such lines all lie on a surface, for we cannot draw one
through an arbitrary point; only one line passes through a point of
the surface; the simplest example, that of a quadric surface, is
really two skews on the same surface.
The degree of a ruled surface _qua_ line geometry is the number of its
generating lines contained in a linear complex. Now the number which
meets a given line is the degree of the surface _qua_ point geometry,
and as the lines meeting a given line form a particular case of linear
complex, it follows that the degree is the same from whichever point
of view we regard it. The lines common to three complexes of degrees,
n1n2n3, form a ruled surface of degree 2n1n2n3; but not every ruled
surface is the complete intersection of three complexes.
Linear complex.
In the case of a complex of the first degree (or linear complex) the
lines through a fixed point lie in a plane called the polar plane or
nul-plane of that point, and those lying in a fixed plane pass through
a point called the nul-point or pole of the plane. If the nul-plane of
A pass through B, then the nul-plane of B will pass through A; the
nul-planes of all points on one line l1 pass through another line l2.
The relation between l1 and l2 is reciprocal; any line of the complex
that meets one will also meet the other, and every line meeting both
belongs to the complex. They are called conjugate or polar lines with
respect to the complex. On these principles can be founded a theory of
reciprocation with respect to a linear complex.
This may be aptly illustrated by an elegant example due to A. Voss.
Since a twisted cubic can be made to satisfy twelve conditions, it
might be supposed that a finite number could be drawn to touch four
given lines, but this is not the case. For, suppose one such can be
drawn, then its reciprocal with respect to any linear complex
containing the four lines is a curve of the third class, i.e. another
twisted cubic, touching the same four lines, which are unaltered in
the process of reciprocation; as there is an infinite number of
complexes containing the four lines, there is an infinite number of
cubics touching the four lines, and the problem is poristic.
The following are some geometrical constructions relating to the
unique linear complex that can be drawn to contain five arbitrary
lines:
To construct the nul-plane of any point O, we observe that the two
lines which meet any four of the given five are conjugate lines of the
complex, and the line drawn through O to meet them is therefore a ray
of the complex; similarly, by choosing another four we can find
another ray through O: these rays lie in the nul-plane, and there is
clearly a result involved that the five lines so obtained all lie in
one plane. A reciprocal construction will enable us to find the
nul-point of any plane. Proceeding now to the metrical properties and
the statical and dynamical applications, we remark that there is just
one line such that the nul-plane of any point on it is perpendicular
to it. This is called the central axis; if d be the shortest distance,
[theta] the angle between it and a ray of the complex, then d tan
[theta] = p, where p is a constant called the pitch or parameter. Any
system of forces can be reduced to a force R along a certain line, and
a couple G perpendicular to that line; the lines of nul-moment for the
system form a linear complex of which the given line is the central
axis and the quotient G/R is the pitch. Any motion of a rigid body can
be reduced to a screw motion about a certain line, i.e. to an angular
velocity [omega] about that line combined with a linear velocity u
along the line. The plane drawn through any point perpendicular to the
direction of its motion is its nul-plane with respect to a linear
complex having this line for central axis, and the quotient u/[omega]
for pitch (cf. Sir R.S. Ball, _Theory of Screws_).
The following are some properties of a configuration of two linear
complexes:
The lines common to the two-complexes also belong to an infinite
number of linear complexes, of which two reduce to single straight
lines. These two lines are conjugate lines with respect to each of the
complexes, but they may coincide, and then some simple modifications
are required. The locus of the central axis of this system of
complexes is a surface of the third degree called the cylindroid,
which plays a leading part in the theory of screws as developed
synthetically by Ball. Since a linear complex has an invariant of the
second degree in its coefficients, it follows that two linear
complexes have a lineo-linear invariant. This invariant is
fundamental: if the complexes be both straight lines, its vanishing is
the condition of their intersection as given above; if only one of
them be a straight line, its vanishing is the condition that this line
should belong to the other complex. When it vanishes for any two
complexes they are said to be in _involution_ or _apolar_; the
nul-points P, Q of any plane then divide harmonically the points in
which the plane meets the common conjugate lines, and each complex is
its own reciprocal with respect to the other. As regards a
configuration of these linear complexes, the common lines from one
system of generators of a quadric, and the doubly infinite system of
complexes containing the common lines, include an infinite number of
straight lines which form the other system of generators of the same
quadric.
General line coordinates.
If the equation of a linear complex is Al + Bm + Cn + D[lambda] +
E[mu] + F[nu] = 0, then for a line not belonging to the complex we may
regard the expression on the left-hand side as a multiple of the
moment of the line with respect to the complex, the word moment being
used in the statical sense; and we infer that when the coordinates are
replaced by linear functions of themselves the new coordinates are
multiples of the moments of the line with respect to six fixed
complexes. The essential features of this coordinate system are the
same as those of the original one, viz. there are six coordinates
connected by a quadratic equation, but this relation has in general a
different form. By suitable choice of the six fundamental complexes,
as they may be called, this connecting relation may be brought into
other simple forms of which we mention two: (i.) When the six are
mutually in involution it can be reduced to x1^2 + x2^2 + x3^2 + x4^2
+ x5^2 + x6^2 = 0; (ii.) When the first four are in involution and the
other two are the lines common to the first four it is x1^2 + x2^2 +
x3^2 + x4^2 - 2x5x6 = 0. These generalized coordinates might be
explained without reference to actual magnitude, just as homogeneous
point coordinates can be; the essential remark is that the equation of
any coordinate to zero represents a linear complex, a point of view
which includes our original system, for the equation of a coordinate
to zero represents all the lines meeting an edge of the fundamental
tetrahedron.
The system of coordinates referred to six complexes mutually in
involution was introduced by Felix Klein, and in many cases is more
useful than that derived directly from point coordinates; e.g. in the
discussion of quadratic complexes: by means of it Klein has developed
an analogy between line geometry and the geometry of spheres as
treated by G. Darboux and others. In fact, in that geometry a point is
represented by _five_ coordinates, connected by a relation of the same
type as the one just mentioned when the five fundamental spheres are
mutually at right angles and the equation of a sphere is of the first
degree. Extending this to four dimensions of space, we obtain an exact
analogue of line geometry, in which (i.) a point corresponds to a
line; (ii.) a linear complex to a hypersphere; (iii.) two linear
complexes in involution to two orthogonal hyperspheres; (iv.) a linear
complex and two conjugate lines to a hypersphere and two inverse
points. Many results may be obtained by this principle, and more still
are suggested by trying to extend the properties of circles to spheres
in three and four dimensions. Thus the elementary theorem, that, given
four lines, the circles circumscribed to the four triangles formed by
them are concurrent, may be extended to six hyperplanes in four
dimensions; and then we can derive a result in line geometry by
translating the inverse of this theorem. Again, just as there is an
infinite number of spheres touching a surface at a given point, two of
them having contact of a closer nature, so there is an infinite number
of linear complexes touching a non-linear complex at a given line, and
_three_ of these have contact of a closer nature (cf. Klein, _Math.
Ann._ v.).
Sophus Lie has pointed out a different analogy with sphere geometry.
Suppose, in fact, that the equation of a sphere of radius r is
x^2 + y^2 + z^2 + 2ax + 2by + 2cz + d = 0,
so that r^2 = a^2 + b^2 + c^2 - d; then introducing the quantity e to
make this equation homogeneous, we may regard the sphere as given by
the six coordinates a, b, c, d, e, r connected by the equation a^2 +
b^2 + c^2 - r^2 - de = 0, and it is easy to see that two spheres
touch, if the polar form 2aa1 + 2bb1 + 2cc1 - 2rr1 - de1 - d1e
vanishes. Comparing this with the equation x1^2 + x2^2 + x3^2 + x4^2 -
2x5x6 = 0 given above, it appears that this sphere geometry and line
geometry are identical, for we may write a = x1, b = x2, c = x3, r =
x4(/[delta] - 1), d = x5, e = 1/2x6; but it is to be noticed that a
sphere is really replaced by two lines whose coordinates only differ
in the sign of x4, so that they are polar lines with respect to the
complex x4 = 0. Two spheres which touch correspond to two lines which
intersect, or more accurately to two pairs of lines (p, p') and (q,
q'), of which the pairs (p, q) and (p', q') both intersect. By this
means the problem of describing a sphere to touch four given spheres
is reduced to that of drawing a pair of lines (t, t') (of which t
intersects one line of the four pairs (pp'), (qq'), (rr'), (ss'), and
t' intersects the remaining four). We may, however, ignore the
accented letters in translating theorems, for a configuration of lines
and its polar with respect to a linear complex have the same
projective properties. In Lie's transformation a linear complex
corresponds to the totality of spheres cutting a given sphere at a
given angle. A most remarkable result is that lines of curvature in
the sphere geometry become asymptotic lines in the line geometry.
Some of the principles of line geometry may be brought into clearer
light by admitting the ideas of space of four and five dimensions.
Thus, regarding the coordinates of a line as homogeneous coordinates
in five dimensions, we may say that line geometry is equivalent to
geometry on a quadric surface in five dimensions. A linear complex is
represented by a hyperplane section; and if two such complexes are in
involution, the corresponding hyperplanes are conjugate with respect
to the fundamental quadric. By projecting this quadric
stereographically into space of four dimensions we obtain Klein's
analogy. In the same way geometry in a linear complex is equivalent to
geometry on a quadric in four dimensions; when two lines intersect the
representative points are on the same generator of this quadric.
Stereographic projection, therefore, converts a curve in a linear
complex, i.e. one whose tangents all belong to the complex, into one
whose tangents intersect a fixed conic: when this conic is the
imaginary circle at infinity the curve is what Lie calls a minimal
curve. Curves in a linear complex have been extensively studied. The
osculating plane at any point of such a curve is the nul-plane of the
point with respect to the complex, and points of superosculation
always coincide in pairs at the points of contact of stationary
tangents. When a point of such a curve is given, the osculating plane
is determined, hence all the curves through a given point with the
same tangent have the same torsion.
Non-linear complexes.
The lines through a given point that belong to a complex of the nth
degree lie on a cone of the nth degree: if this cone has a double line
the point is said to be a singular point. Similarly, a plane is said
to be singular when the envelope of the lines in it has a double
tangent. It is very remarkable that the same surface is the locus of
the singular points and the envelope of the singular planes: this
surface is called the singular surface, and both its degree and class
are in general 2n(n - 1)^2, which is equal to four for the quadratic
complex.
The singular lines of a complex F = 0 are the lines common to F and
the complex
[delta]F [delta]F [delta]F [delta]F [delta]F [delta]F
-------- --------------- + -------- ----------- + -------- ----------- = 0.
[delta]l [delta][lambda] [delta]m [delta][mu] [delta]n [delta][nu]
As already mentioned, at each line l of a complex there is an infinite
number of tangent linear complexes, and they all contain the lines
adjacent to l. If now l be a singular line, these complexes all reduce
to straight lines which form a plane pencil containing the line l.
Suppose the vertex of the pencil is A, its plane a, and one of its
lines [xi], then l' being a complex line near l, meets [xi], or more
accurately the mutual moment of l', and is of the second order of
small quantities. If P be a point on l, a line through P quite near l
in the plane a will meet [xi] and is therefore a line of the complex;
hence the complex-cones of all points on l touch a and the
complex-curves of all planes through l touch l at A. It follows that l
is a double line of the complex-cone of A, and a double tangent of the
complex-curve of a. Conversely, a double line of a cone or curve is a
singular line, and a singular line clearly touches the curves of all
planes through it in the same point. Suppose now that the consecutive
line l' is also a singular line, A' being the allied singular point,
a' the singular plane and [xi]' any line of the pencil (A', a') so
that [xi]' is a tangent line at l' to the complex: the mutual moments
of the pairs l', [xi] and l, [xi] are each of the second order; hence
the plane a' meets the lines l and [xi]' in two points very near A.
This being true for all singular planes, near a the point of contact
of a with its envelope is in A, i.e. the locus of singular points is
the same as the envelope of singular planes. Further, when a line
touches a complex it touches the singular surface, for it belongs to a
plane pencil like (Aa), and thus in Klein's analogy the analogue of a
focus of a hyper-surface being a bitangent line of the complex is also
a bitangent line of the singular surface. The theory of cosingular
complexes is thus brought into line with that of confocal surfaces in
four dimensions, and guided by these principles the existence of
cosingular quadratic complexes can easily be established, the analysis
required being almost the same as that invented for confocal cyclides
by Darboux and others. Of cosingular complexes of higher degree
nothing is known.
Following J. Plucker, we give an account of the lines of a quadratic
complex that meet a given line.
The cones whose vertices are on the given line all pass through eight
fixed points and envelop a surface of the fourth degree; the conics
whose planes contain the given line all lie on a surface of the fourth
class and touch eight fixed planes. It is easy to see by elementary
geometry that these two surfaces are identical. Further, the given
line contains four singular points A1, A2, A3, A4, and the planes into
which their cones degenerate are the eight common tangent planes
mentioned above; similarly, there are four singular planes, a1, a2,
a3, a4, through the line, and the eight points into which their conics
degenerate are the eight common points above. The locus of the pole of
the line with respect to all the conics in planes through it is a
straight line called the _polar line_ of the given one; and through
this line passes the polar plane of the given line with respect to
each of the cones. The name polar is applied in the ordinary
analytical sense; any line has an infinite number of polar complexes
with respect to the given complex, for the equation of the latter can
be written in an infinite number of ways; one of these polars is a
straight line, and is the polar line already introduced. The surface
on which lie all the conics through a line l is called the Plucker
surface of that line: from the known properties of (2, 2)
correspondences it can be shown that the Plucker surface of l cuts l1
in a range of the same cross ratio as that of the range in which the
Plucker surface of l1 cuts l. Applying this to the case in which l1 is
the polar of l, we find that the cross ratios of (A1, A2, A3, A4) and
(a1, a2, a3, a4) are equal. The identity of the locus of the A's with
the envelope of the a's follows at once; moreover, a line meets the
singular surface in four points having the same cross ratio as that of
the four tangent planes drawn through the line to touch the surface.
The Plucker surface has eight nodes, eight singular tangent planes,
and is a double line. The relation between a line and its polar line
is not a reciprocal one with respect to the complex; but W. Stahl has
pointed out that the relation is reciprocal as far as the singular
surface is concerned.
Quadratic complexes.
To facilitate the discussion of the general quadratic complex we
introduce Klein's canonical form. We have, in fact, to deal with two
quadratic equations in six variables; and by suitable linear
transformations these can be reduced to the form
a1x1^2 + a2x2^2 + a3x3^2 + a4x4^2 + a5x5^2 + a6x6^2 = 0
x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 = 0
subject to certain exceptions, which will be mentioned later.
Taking the first equation to be that of the complex, we remark that
both equations are unaltered by changing the sign of any coordinate;
the geometrical meaning of this is, that the quadratic complex is its
own reciprocal with respect to each of the six fundamental complexes,
for changing the sign of a coordinate is equivalent to taking the
polar of a line with respect to the corresponding fundamental complex.
It is easy to establish the existence of six systems of bitangent
linear complexes, for the complex l1x1 + l2x2 + l3x3 + l4x4 + l5x5 +
l6x6 = 0 is a bitangent when
l2^2 l3^2 l4^2 l5^2 l6^2
l1 = 0, and ------- + ------- + ------- + ------- + ------- = 0
a2 - a1 a3 - a1 a4 - a1 a5 - a1 a6 - a1
and its lines of contact are conjugate lines with respect to the first
fundamental complex. We therefore infer the existence of six systems
of bitangent lines of the complex, of which the first is given by
x2^2 x3^2 x4^2 x5^2 x6^2
x1 = 0, ------- + ------- + ------- + ------- + ------- = 0.
a2 - a1 a3 - a1 a4 - a1 a5 - a1 a6 - a1
Each of these lines is a bitangent of the singular surface, which is
therefore completely determined as being the focal surface of the (2,
2) congruence above. It is thence easy to verify that the two
complexes [Sigma]ax^2 = 0 and [Sigma]bx^2 = 0 are cosingular if b_r =
a_r[lambda] + [mu]/a_r[nu] + [rho].
The singular surface of the general quadratic complex is the famous
quartic, with sixteen nodes and sixteen singular tangent planes, first
discovered by E.E. Kummer.
We cannot give a full account of its properties here, but we deduce at
once from the above that its bitangents break up into six (2, 2)
congruences, and the six linear complexes containing these are
mutually in involution. The nodes of the singular surface are points
whose complex cones are coincident planes, and the complex conic in a
singular tangent plane consists of two coincident points. This
configuration of sixteen points and planes has many interesting
properties; thus each plane contains six points which lie on a conic,
while through each point there pass six planes which touch a quadric
cone. In many respects the Kummer quartic plays a part in three
dimensions analogous to the general quartic curve in two; it further
gives a natural representation of certain relations between
hyperelliptic functions (cf. R.W.H.T. Hudson, _Kummer's Quartic_,
1905).
Classification of quadratic complexes.
As might be expected from the magnitude of a form in six variables,
the number of projectivally distinct varieties of quadratic complexes
is very great; and in fact Adolf Weiler, by whom the question was
first systematically studied on lines indicated by Klein, enumerated
no fewer than forty-nine different types. But the principle of the
classification is so important, and withal so simple, that we give a
brief sketch which indicates its essential features.
We have practically to study the intersection of two quadrics F and F'
in six variables, and to classify the different cases arising we make
use of the results of Karl Weierstrass on the equivalence conditions
of two pairs of quadratics. As far as at present required, they are as
follows: Suppose that the factorized form of the determinantal
equation Disct (F + [lambda]F') = 0 is
([lambda] - [alpha])^(s1 + s2 + s3 ...)
([lambda] - [beta])^(t1 + t2 + t3 + ...) ...